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Lecture Comments (2)

1 answer

Last reply by: Professor Selhorst-Jones
Fri Mar 25, 2016 5:12 PM

Post by Jay Lee on March 21 at 06:28:28 AM


How would you simplify (x^(5/4)y^(-5/4))/(x^(-1/2)y^(7/4)x^(2)y^(2))^(5/3)? The question requires the answer to contain only positive exponents with no fractional exponents in the denominator. I tried solving the question and ended up with a fraction in the denominator, so I changed those to the "root" form, but the teacher wouldn't accept that answer either. I solved until: (y^(1/2)x^(3/4))/(y^(8)x^(2)), but I don't know how to simplify it further.

Thank you so much:)

Exponential Functions

  • An exponential function is a function of the form
    f(x)   =   ax,
    where x is any real number and a is a real number such that a ≠ 1 and a > 0. We call a the base. [The base is the thing being raised to some exponent.]
  • From the previous lesson, Understanding Exponents, we can compute the value of a given base raised to any exponent. In practice, we can find these expressions (or a very good approximation) by using a calculator. Any scientific or graphing calculator can do such calculations.
  • Exponential functions grow really, really, REALLY fast. There's a fictional story in the lesson to get across just how incredibly fast these things grow. Multiplying by a factor repeatedly can get to extremely large values in a short period of time.
  • Compound interest is a form of interest on an investment where the interest gained off the principal (the amount of money initially invested) also gains interest. Thus, compound interest returns more and more money the longer the investment is left in.
  • We can describe the amount of money, A, in a compound interest account with an exponential function:
    A(t)   =  P
    1 + r



    • P is the principal in the account-the amount originally placed in the account,
    • r is the annual rate of interest (given as a decimal: 6% ⇒ 0.06),
    • n is the number of times a year that the interest compounds (n=1⇒ yearly; n=4⇒ quarterly; n=12 ⇒ monthly;  n=365 ⇒ daily),
    • t is the number of years elapsed.
  • In the above equation, we can see that the more often the interest compounds, the more money we make. This motivates us to create a new number: the natural base, e. (See the video for more information on how we find e.) The number e goes on forever:
    e = 2.  718  281  828  459  045  235  360 …
    Like π, the number e is an irrational number: its decimal expansion continues forever, never repeating. Also like π, the number e is deeply connected to some fundamental things in math and the nature of the universe. Just as π is fundamentally connected to how circles work, the number e is fundamentally connected to how things that are continuously growing work.
  • One application of e is to see how an account would grow if it was being compounded every single instant- being continuously compounded:
    A(t) = Pert,
    • P is the principal (starting amount),
    • r is the annual rate of interest (given as a decimal),
    • t is the number of years elapsed.
    The above equation can also be used for a wide variety of things that grow (or decay) continuously. You'll see it show up a lot in math and science.
  • We can see exponential decay if 0 < a < 1 in f(x) = ax. The exponential function will quickly become very small as it repeatedly "loses" its value because of the fraction (a) being compounded over and over.

Exponential Functions

Find the value of e3 to three decimal places.
  • The symbol e represents a number in the same way that π does. Also, just like π, e is irrational: its decimal expansion never ends and never repeats.
    e = 2.  718  281  828…
  • Clearly, we can't use all the digits of e because there are infinitely many of them. However, the more of them we use, the more accurate our calculation will be. Since we need three decimal places of accuracy, let's begin by using at least six decimals from e: 2.  718  281. [This won't be perfectly accurate, but it will be accurate enough to find e3 to three decimal places. If we want more accuracy, we can use more decimals from e.]
  • Plug the approximation into a calculator:
    e3     ≈     2.7182813 = 20.085  518  558  …
    Thus, we round to three places, saying e3 ≈ 20.086. [Notice that the above approximation is not perfect. As we use more decimals, we get a more accurate result. Using a massive number of digits, we could find that
    e3 = 20.085  536  923  …
    See how the highly accurate value for e3 above starts to give different values after the fourth digit. While we can get good accuracy with a modest approximation of e, if we really need a massive amount of accuracy, we have to start with a lot of accuracy in the value we use for e.]
20.086 [If your answer is close, but off by the last couple digits, you probably need to use more digits of e. Look at the steps for a detailed explanation.]
Complete the table below for f(x) = 3x.
  • To fill in the table, plug in each value to the function.
  • For 3−1, remember that raising anything to the negative exponent causes it to "flip" to its reciprocal.
  • For 30, remember that raising anything to the 0 causes it to become 1.
  • As you fill out the table, notice how very quickly the function grows. Exponential functions become massively huge very, very quickly.


The function f(x) is an exponential function. Using the table of values below, state the function f(x).
  • An exponential function is a function of the form f(x) = ax, where a is some base. Thus, to answer the question, we need to figure out what a is.
  • The fact that f(0)=1 is not much help to us. As long as the function is strictly in the form ax, then any value of a will still produce 1. This is because raising any number to the 0 causes it to become 1. So we can't get much useful information from the first entry in the table.
  • The second and third entries are much more useful, though. We know f(2) = a2 = 22.09 and f(3) = a3 = 103.823. Notice that a2 ·a = a3. Thus,
    a2 ·a = a3     ⇒     22.09 ·a = 103.823     ⇒     a = 4.7
  • Finally, now that we know a=4.7, we can double-check our answer. Currently, we have f(x) = 4.7x. From the table, we know f(4) should be 487.9681. Check using a calculator that 4.74 gives the same value. Indeed, it does, so our answer is correct.
f(x) = 4.7x
Graph the function f(x) = 4x−2−17.
  • Like graphing any function, we can always use brute force: calculate and plot a bunch of points, then draw a graph from those points. Approaching it that way, we might create a table such as the below:
  • Alternatively, we could examine the structure of the function. Notice that f(x) = 4x−2−17 is similar to the expression 4x. From looking at graphs of other exponential functions, it might be very easy for us to graph 4x: it would be 1 at x=0, 4 at x=1, and shoot up rapidly as we go farther to the right. If we examine 4x as it goes to the left, it would approach the x-axis asymptotically. Now notice that 4x−2 would have the same graph as 4x, except it would be shifted 2 units to the right. Next, 4x−2−17 would be shifted a further 17 units down. [If you're not sure how to see this, check out the lesson on function transformations.] Thus, f(x) = 4x−2 −17 will have the same shape as 4x, but shifted 2 units right and 17 units down.
  • Either way that we approach understanding the shape of f(x), we now need to graph it. Notice that f(x) moves very quickly vertically for a small amount horizontally. Thus, we might want to choose graphing axes that are very tall for a small width.
Graph the function g(x) = ( [2/3] )x.
  • Because the base of the exponent is smaller than 1, the value of the function will become very small for positive x. As the fractions multiply each other more and more, the result will be tiny. On the other hand, when we have a negative x, the negative exponent will cause the fraction to flip, and thus allow the function to grow.
  • To help us see how the function graphs, calculate various points:
  • Plot points and connect with curves. Notice that as x moves very far to the right, the function will asymptotically approach a value of 0. On the other hand, as x moves to the left, the function will grow very rapidly.
A bank account is opened and $10 000 is placed in the account. If the account has an interest rate of 4.7%, how much money is in the account after 25 years if the account compounds annually? What if it had been compounded semiannually? Daily?
  • We can describe the amount of money, A, in such an account with an exponential function:
    A(t)   =  P
    1 + r



    where P is the principal in the account, r is the interest rate (as a decimal), n is the number of times the account compounds per year, and t is the number of years elapsed.
  • For this problem, we have
    P=10 000,        r = 0.047,        t = 25.
    The only thing left is how many times the account compounds per year, and the question asks us to figure that out for three different compounding schedules.
  • Those compounding schedules are as follows: annually-once a year ⇒ n=1; semiannually-twice a year ⇒ n=2; daily-every day, so 365 times in a year ⇒ n=365. Use this along with the information above to figure out how much money is in the account for each situation.

  • Annually:     10000
    1 + 0.047



        =     10000 (1.047)25     =     31 525.87

    Semiannually:     10000
    1 + 0.047



        =     10000 (1.0235)50     =     31 944.22

    Daily:     10000
    1 + 0.047



        =     10000 (1.000128767…)9125     =     32 378.98
Annually: $31 525.87,    Semiannually: $31 944.22,    Daily: $32 378.98
Sally is currently 45 years old and is planning to retire at age 65. She has $25 000 in savings that she is going to put into an investment that has an interest rate of 7% and compounds continuously. How much money will she have when she retires? How much money would she have at retirement if she had made the same investment, but had instead done it at age 35? What about age 25?
  • If something is being compounded continuously, the amount (A) of money after t years is
    A(t) = Pert,
    where P is the principal, r is the interest rate (in decimal), and e is the natural base.
  • The principal investment is P=25 000 and the rate is r=0.07. If she puts the money into the investment at age 45, then retires at age 65, that means the investment has t=20 years to mature. Thus, its value is
    25000 e0.07 ·20     =     25000 e1.4     =     101 380.00
  • For the other ages, we just need to figure out new values for t. If she invests at age 35, then that means t=30 years until retirement:
    25000 e0.07 ·30     =     25000 e2.1     =     204 154.25
    If she makes the investment at 25, then she has t=40 years:
    25000 e0.07 ·40     =     25000 e2.8     =     411  116.17
    [Notice the massive increase in the value of the investment. Because compounding investments are based on exponential functions, the earlier the investment can be made, the greater its final value will wind up being.]
Age 45: $101 380.00,    Age 35: $204 154.25,     Age 25: $411  116.17
Bryan buys a new car for $26 650. Because of depreciation, the value of the car goes down with a time. The value of the car after t years is
p(t) = 26 650 (0.87)t.
What is the value of the car after one year of use? If Bryan sells the car after using it for a total of seven years, what will have been his net cost in owning the car? [Hint: The net cost is the original cost of the car, minus the amount he gets when selling the car.]
  • We have a formula to determine the value of the car after t years of use. To find the value of the car after one year of use, simply plug in t=1:
    p(1) = 26650 (0.87)1     =     23 185.50
  • To find his net cost upon selling the car at seven years, we first need to know how much he sells the car for. Figure out the value of the car after seven (t=7) years:
    p(7) = 26650 (0.87)7     =     10 053.84
  • Now that we know Bryan is able to sell the car for $10 053.84, we can calculate his net cost in owning the car. While it cost him $26 650 to purchase the car in the first place, he ultimately managed to sell it for some money, thus lowering the net cost of owning the car. To find the net cost, simply subtract the selling price from the original cost:
    26650−10053.84     =     16 596.16
Value of car after one year: $23 185.50,    Net cost of owning for seven years: $16 596.16
A population of bacteria doubles every 26 minutes. If the population starts with 50 cells, how many will there be in 12 hours? (Assume that none of the cells die off.)
  • Notice that we are told the population doubles after a certain time interval. This is equivalent to multiplying the starting population by 2 after a time interval passes. After the next time interval, it will multiply by 2 again, and so on. Thus, if we say the number of time intervals is n, and the starting population is P, the amount A of bacteria will be
    A = P·2n.
  • Now we need to figure out how many time intervals have elapsed. We know that one interval occurs every 26 minutes. The total time is 12 hours, which is 720 minutes. Thus, the number of intervals that have occurred is
    n = 720

  • We can plug that in to the above to figure out the amount of bacteria after 12 hours:
    A = 50 ·2[720/26]     =     10 843 894 130
10 843 894 130 bacteria
The radioactive isotope Carbon-14 has a half-life of 5 730 years. (Half-life is the time it takes for [1/2] of the isotope to decay and break down.) If we start with a quantity of 2.3  μg (micrograms) of C-14, how much will have not decayed after 30 000 years?
  • Notice that we are told the quantity halves after a certain time interval. This is equivalent to multiplying the starting amount by [1/2] after a time interval passes. After the next time interval passes, it will multiply by [1/2] again, and so on. Thus, if we say the number of time intervals is n, and the starting quantity is P, the amount A of quantity in the end will be
    A = P·



  • Now we need to figure out how many time intervals have elapsed. We know that one interval occurs every 5 730 years. The total time is 30 000 years. Thus, the number of intervals that have occurred is
    n = 30 000

    5 730
  • We can plug that in to the above to figure out the quantity of not decayed C-14 after the passage of 30 000 years:
    A = 2.3 ·



        =     0.061
0.061 μg

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.


Exponential Functions

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Introduction 0:05
  • Definition of an Exponential Function 0:48
    • Definition of the Base
    • Restrictions on the Base
  • Computing Exponential Functions 2:29
    • Harder Computations
    • When to Use a Calculator
  • Graphing Exponential Functions: a>1 6:02
    • Three Examples
    • What to Notice on the Graph
  • A Story 8:27
    • Story Diagram
    • Increasing Exponentials
    • Story Morals
  • Application: Compound Interest 15:15
    • Compounding Year after Year
    • Function for Compounding Interest
  • A Special Number: e 20:55
    • Expression for e
    • Where e stabilizes
  • Application: Continuously Compounded Interest 24:07
    • Equation for Continuous Compounding
  • Exponential Decay 0<a<1 25:50
    • Three Examples
    • Why they 'lose' value
  • Example 1 27:47
  • Example 2 33:11
  • Example 3 36:34
  • Example 4 41:28

Transcription: Exponential Functions

Hi--welcome back to

Today, we are going to talk about exponential functions.0002

Previously, we spent quite a while looking at functions that are based around a variable raised to a number--0005

things like x2 or x47; this is basically the idea of all of those polynomials we have worked with for so long.0010

But what if we took that idea and flipped it?0017

We could consider functions that are a number raised to a variable, things like 2x or 47t,0019

where we have some base number that has a variable as its exponent.0027

We call functions of this form exponential functions, and we will explore them in this lesson.0032

Now, make sure that you have a strong grasp on how exponents work before watching this.0036

If you need a refresher on how exponents work, check out the previous lesson, Understanding Exponents, to get a good grounding in how exponents work.0040

All right, an exponential function is a function in the form f(x) = ax,0048

where x is any real number, and a is a real number such that a is not equal to 1, and a is greater than 0.0054

We call a the base: base is just the name for the thing that is being raised to some exponent.0060

So, whatever is being exponentiated--whatever is going through this process of having an exponent--that is called the base,0066

because it forms the base, because it is below the exponent.0072

We might wonder why there are all of these restrictions on what a can be; well, there are good reasons for each one.0076

If a equals 1, we just have this boring constant function, because we would have 1x, which is just equal to 1 all of the time.0081

So, something that is just equal to 1 all of the time is not really interesting, and it is not really going to be an exponential function.0091

So, we are not going to consider that case.0097

If a equals 0, the function wouldn't be defined for negative values of x.0100

If we try to consider what 0-1 is, well, then we would get 1/0, but we can't do that--we can't divide by 0.0104

So, that is not allowed, so that means a = 0--once again, we are not going to allow that one.0111

And if we had a < 0, then the function wouldn't be defined for various x-values, like x1/2.0116

For example, if we had -4, and we raised that to the 1/2, well, we know that raising it to the 1/2 is the same thing as taking the square root.0121

So, the square root of -4...we can't take the square root of a negative, because that produces imaginary numbers;0129

and we are only dealing with real numbers--we are not dealing with the complex numbers right now.0134

So, we are going to have to ban anything that is less than 0.0137

And that is why we have this restrictions: our base has to be greater than 0, and is not allowed to be 1,0140

because otherwise things break down for the exponential function.0145

All right, notice that, from the previous lesson, we can compute the value of a given base raised to any exponent.0149

We know how exponents work when they are a little more complex (not complex numbers, but just more interesting).0154

And so, we can raise things like...43/2 = (√4)3,0160

which would be equal to...√4 is 2; 23, 2 times 2 times 2, gives us 8; great.0165

If we had 7-2, well, then that would become 1/7, because we have the negative, so the negative flips it to (1/7)2.0171

So, 1 squared is 1; 7 squared is 49; we get 1/49.0179

So, we can do these things that are a little more difficult than just straight positive integers.0184

But we might still find some calculations difficult, like if we had 1.76.2--that would probably be pretty hard to do.0190

Or (√2)π--these would be really difficult for us to do.0196

So, how do we do them? In practice, we just find these expressions, or a very good approximation, by using a calculator.0200

We can end up getting as many digits in our decimal expansion as we want.0206

We can just find as many as we need for whatever our application is--whatever the problem asks for--by just using a calculator.0214

Any scientific or graphing calculator can do these sorts of calculations.0220

There will be some little button that will say xy, or some sort of _ to the _--some way to raise to some other thing--something random.0223

They might have a carat, which says...if I have 36 (not with an a--I accidentally drew that in...oh, I drew it in again),0235

then that would be equivalent to us saying 36.0250

The carat is saying "go up," so the calculator would interpret 36 as 36.0253

There are various ways, depending on if you are using a scientific calculator,0259

or if you are using a graphing calculator, to put these things into a calculator and get a number out.0261

So, we are able to figure these things out, just by being able to say "use a calculator."0266

Now, from a mathematical point of view, that is a terrible statement.0270

We don't want to say, "We can deal with this because we have calculators!" because how did you figure it out before you had calculators?0273

Calculators didn't just spawn into existence and give us the answers.0279

We can't rely on our calculators to do our thinking for us; we have to be able to understand what is going on.0283

Otherwise, we don't really have a clue how it works.0287

But as you will see as you get into more advanced math classes, there are methods to figure out these values.0290

There are ways to do this by hand, because there are various algorithms that give us step-by-step ways to get a few decimals at a time.0296

Now, doing it by hand is long, slow, and tedious; it would be hard to get this sort of thing, just because it would be so much calculation to do.0303

We could do it, but that is what calculators are for; they are to do lots of calculations very quickly.0310

They are to help us get through tedious arithmetic.0316

So, since these sorts of calculations take all of this arithmetic, we designed calculators that can do this method for us.0319

And that is why we can appeal to a calculator--not because the calculator knows more than us,0326

but because, at some point, humans figured out a method to get as many decimals as we wanted to;0331

and then, we just built a machine that is able to go through it quickly and rapidly,0336

so we can get to the thing that we want to look at, which is more interesting, using this.0340

The calculator is a tool; but it is important to realize that we are not just relying on it because it has the knowledge.0345

We are relying on it because, at some point, we built it and put these methods into it.0350

And if you keep going in mathematics, you will eventually see that these are where the methods come from--there is some pretty interesting stuff in calculus.0354

All right, now, if we can evaluate at any place--if we can compute what these values of exponential functions are--0361

then we can make a graph, because we can plot as many points as we want; we can draw a smooth curve.0368

So, let's look at some graphs where the base is greater than 1.0372

If we have 2x, that would be the one in red; 5x is the one in blue, and 10x is the one in green.0375

Now, notice: 2x, 5x, and 10x--all of these end up going through 1, right here,0384

because what is happening there is that 20, 50, 100...anything raised to the 0--they all end up being 1.0392

Remember, that is one of the basic properties of exponents.0401

If you raise something to the 0, it just becomes 1; so that is why we see all of them going through the same point.0403

And notice that they get very large very quickly.0409

By the time 2 is to the fourth, it is already off; and 10 is off by the time it gets to the 1.0411

10x grows very quickly, because it is multiplying by 10, each step it goes forward.0417

Notice also: as we go far to the left, it shrinks very quickly.0421

Let's consider 10-3; 10-3 would be the same thing as 1/103, which would be 1/1000.0425

That is why we end up seeing that this green line is so low.0436

It looks like it is almost touching the x-axis; it isn't quite--there is this thin sliver between it.0440

But it is being crushed down very, very quickly, because of this negative exponent effect,0444

where it gets flipped over, and then it has a really, really large denominator very quickly.0448

So, we see, as we go to the left side with these things, that it will crush down to 0.0454

And as we go to the right, it becomes very, very big.0460

We can change the viewing window, so that we can get a sense for just how big these things get.0464

And look at how big: we have gotten up to the size of 1000 by the time we are only out to 10.0468

And that is on 2x; if we look at 10x, 10x has already hit 1000 at 103.0475

At x = 3, it has managed to hit 1000 as its height.0483

This stuff grows really quickly; this idea of massive growth is so central to the idea of exponential functions.0488

We are going to have a story: there is this story that often gets told with exponential functions,0494

because it is a great way to get people to understand just how big this stuff gets.0499

So, let's check it out: All right, long ago, in a far-off land, there was a mathematician who invented the game of chess.0505

The king of the land loved the game of chess so much that he offered the mathematician any reward that the mathematician desired.0513

The mathematician was clever, and told the king humbly, "Your Highness, I thank you;0519

all I ask for is a meager gift of rice, given day by day on a chessboard."0523

"Tomorrow, I would like a single grain of rice give on the first square;0528

on the next day, two grains of rice given on the second square; then on the following day, the third day,0533

four grains of rice; and so on and so forth, doubling the amount every day until all 64 squares are filled."0541

So, the mathematician is asking for the first square, doubled, doubled, doubled, doubled...0550

The mathematician drew the king a diagram to help make his request clear.0556

On the first day of his gift, he would end up having one grain of rice on the first square.0559

On the second day, there would be a total of two grains of rice (1 times 2 becomes 2).0567

On the third day, there would be a total of 4 grains of rice (2 times 2 becomes 4).0572

On the next day, there will be 8 (4 times 2 becomes 8), and then 16, and then 32, and so on and so on and so on,0578

going all the way out to the 64th day, doubling each time we go forward a square on the board.0586

The king was delighted by the humble request and agreed to it immediately.0593

Grains of rice? You can't get a lot of grains of rice on a single chessboard; "It will be very easy," he thought.0597

He ordered that the mathematician would have his daily reward of rice delivered from the royal treasury every day.0603

A week later, the king marveled at how the mathematician had squandered his reward.0609

After all, he only had to send him 26 = 64 grains of rice that day.0612

Notice: on the seventh day, we are at 26--let's see why that is.0617

On the first day, we have 1 grain; on the second day, we have 2 grains; on the third day, we have 4 grains.0624

On the fourth day, we have 8 grains; on the fifth day, we have 16 grains; on the sixth day, we have 32 grains.0631

And thus, on the seventh day, we have 64 grains.0638

So, notice: we can express this as 20, 21, 22, 23, 24, 25, and then finally 26 on the seventh day.0642

Why is this? Because on the very first day, he just got one grain.0655

Every following day, it multiplies by 2--it doubles.0659

So, that means we multiply it by 2; so we count all of the days after the first day, which is why, on the seventh day, we see an exponent of 6.0662

So, in general, it is going to be 2 to the (number of day minus 1).0671

We will subtract one to figure out the grains on some number of day.0678

So now, we have an idea of how we can calculate this pretty quickly and be able to get these things figured out.0683

Another week later, on the fourteenth day, the king sent him 213 (remember, it is the fourteenth day,0689

so we go back one, because it has been multiplied 13 times) grains of rice, which is 8,192 grains.0694

And 8,192 grains is just about a very large bowl of rice.0702

The king was still amazed at the fantastic deal he was getting.0706

But he was glad that the mathematician was at least seeing some small reward.0709

He loved the game of chess, after all; and if he ended up feeding the mathematician for a year, that was great.0713

It seemed like a wonderful deal; he was willing to give him palaces, jewels, and massive amounts of money.0719

He can give him a little bit of rice for the great game of chess.0724

At the end of the third week, on the twenty-first day, the king had to send the mathematician a full bag of rice,0728

because in the kingdom, a full bag of rice contained precisely 1 million grains.0734

So, on the 21st day, we have 220 grains of rice, which ends up being 1 million, 48 thousand, 576 grains.0739

So we see here: after we jump these first six digits, we have one million plus grains of rice.0747

So, he has managed to get one million grains of rice (which is one bag of rice), plus an extra 48000 in change, in grains.0753

So, perhaps the mathematician was not as foolish as the king had first thought.0761

At the end of the fourth week, the king was starting to get worried.0766

On the twenty-eighth day, he had to send him more than 134 bags of rice, because 227 is more than 134 million grains of rice.0769

So, we are starting to get to some pretty large amounts here.0780

Now, the royal treasury has a lot of rice; he is not worried--he has hundreds of thousands of bags of rice.0782

So, he is not too worried about it; but he sees that this is starting to grow quite a bit.0788

At that moment, the royal accountant bursts into the throne room and says, "Your Highness, I have grave news!0793

The mathematician will deplete the royal treasury! On the forty-first day alone, we would have to give one million bags of rice!"0799

because 240 is here, so we have one million million grains of rice; so we have one million bags of rice,0806

which is more than the entirety that the treasury has in rice.0816

"And if we kept going--if we let it run all the way to the sixty-fourth day,0819

we would have to send him more rice than the total that the world has ever produced,0823

because we would be at 263, which would come out to be 9 trillion bags of rice."0827

Look; we have ones here; we have thousands here; we have millions here, billions here, trillions here, quadrillions here;0835

it would be 9 quintillion grains of rice; if we knock off these first ones, we see that we are still at 9 trillion bags of rice.0842

That is a lot of rice, and the world doesn't have that much by far.0851

So, the mathematician's greed has enraged the king, and the king immediately orders all shipments of rice stopped.0856

The mathematician is not getting any more rice, and the mathematician is to be executed!0862

Now, the mathematician, being a clever fellow, hears the soldiers coming down the road, and he escapes.0866

He fled the kingdom with the few bags of rice that he could manage to carry on his back,0871

and he had to find a new place to live, far, far away from the kingdom.0875

So, the moral of the story is twofold: first, don't be overly greedy--don't try to trick kings.0880

But more importantly than that, exponential functions grow really, really large in a short period of time.0886

They get big fast; even if they start at a seemingly very, very small, miniscule amount, they will grow massive if given enough time.0895

So, that is the real take-away here from this story.0904

Exponential functions get big; they can start small, but given some time, they get really, really big.0907

All right, let's see an application of this stuff.0916

When you put money in a bank, they will usually give you interest on your money.0918

For example, if you had an annual interest rate of 10% (annual just means yearly) on a $100 principal investment0922

(the amount that you put in the bank), the following year you would have that $100 still0931

(they don't take it away from you), plus $100 times 10%.0935

Now, 10% as a decimal is .10; so it is $100 times .10, so you would get that $100 that you originally started with, and you would have $10 in interest.0940

Great; but you could leave that interest in the account, and then your interest would also gain interest.0948

The interest is going to get interest on top of it; so we would say that the interest is compounded, because we are putting on thing on top of the other.0954

So, you have $110 in your bank account now, because you had $110 total at the end last time.0961

$110 gets hit by that 10% again; so you still have the $110, 10% of $110 is $11.0968

Notice that $11 is bigger than 10--your interest is growing.0978

Over time, you are getting more and more interest as you keep letting it stay in there.0985

You continue to gain larger and larger amounts with each interest.0990

Compound interest is a common and excellent way to invest money, because over time,0993

your interest gains interest, and gains interest, and gains interest.0998

And eventually, it can manage to get large enough to be even larger than the principal investment,1002

and be the thing that is really earning you money--the time that you have spent letting it compound.1006

We can describe the amount of money, A, in such an account with an exponential function: A(t) = P[(1 + r)/n]nt.1011

Let's unpack that: P is the principal in the account--the amount that is originally placed in the account.1024

So, in our example, that would be $100 put in; so our principal would be 100 in that last example.1032

r is the annual rate of interest, and we give that as a decimal: here is our r, right up here.1038

In the last one, that was 10%, so it was expressed as .10.1045

n is the number of times a year that the interest compounds; n is the number of times that we see compounding.1050

So, n = 1 would be yearly; n = 4 would be quarterly; n = 12 would be monthly; n = 365 would be daily.1057

In our last one, it compounded annually, every year; so it compounded just once a year, so n was equal to 1.1066

Notice that n also shows up up here; it is n times t.1072

And then finally, t is just the number of years that we have gone through; so it is times t.1077

So, let's understand why this is the case.1084

Well, if we looked at 10%, just on the $100, we would have $100 times 1 + 10%.1086

So, $100 times 1.1 equals $110.1096

Now, if we wanted to have this multiple times, well, the next time it is $110 times 1.1, again.1102

We would get another number out of it; and then, if we wanted to keep hitting it...1109

we can just think of it as (1001.1)t, and that will just give us the amount of times1112

that the interest has hit, over and over and over--our principal times the 11119

(because the bank lets you keep what you started with), plus the interest in decimal form,1125

all raised to the t--the number of years that have elapsed.1131

Now, what about that "divide by n" part?1134

Well, let's say that we compounded it twice a year; so they didn't just give you your interest1136

in a lump sum at the end of the year--they gave it to you in bits and pieces.1140

So, the first time it compounds, if they did it twice a year (let's say they did it semiannually, two times a year),1144

then it would be 1 (because they let you keep the amount of money), plus .1/2 (because they are going to do it twice in a year).1151

So, the first time in the year, we would get 100 times (1 + 0.05), 100 times 1.05.1158

The first time in the year it gets hit, you would get $105 out of that.1169

Now, they could do it again, and we would have $105 get hit with another one of 1.05, and then we could calculate that again.1173

And that would be the total amount that you would have over the year.1181

Now, notice: 105 times 1.05 is going to be a little bit extra, because we are getting that 5 times 1.05, in addition to what we would have ended up having.1183

We would have 105 times 1.05; we will end up getting 5.25 out of this.1192

So, we will have a total of 110 dollars and 25 cents.1200

So, by compounding twice in a year, we end up getting 25 cents more than we did by compounding just once in a year.1208

So, the more times we compound, we get more chances to earn interest on interest on interest.1214

1 + .1, divided by is going to happen twice in a year; so since it happens twice in a year,1219

we have to have the number of times that it is happening in a year, times the number of years.1225

So, at the twice-in-a-year scale, we would see 1.05 to the 2 times number of years, because it happens twice every year.1230

And this method continues the whole time; so that is why we have the divide by n, because the rate has to be split up that many times.1240

But then, it also has to get multiplied that many times extra, because it happens that many times extra in the year.1246

So, that is where we see this whole thing coming from.1252

Now, we noticed, over the course of doing that, that the more times it compounded, the better.1255

We earn more interest if it is calculated more often; the more often our account compounds,1260

the more interest we earn, because we have more chances to earn interest on top of interest.1267

So, we would prefer if it compounded as often as possible--every minute--every second--every instant--1271

if we had it happening continuously--absolutely constantly.1278

This idea of having it happen more and more often leads to the idea of the natural base, which we denote with the letter e.1281

The number e comes from evaluating 1 + 1/n to the n as n approaches infinity--as this becomes larger and larger--1288

because remember: the structure last time was 1 plus this rate, divided by n to the n times t.1295

So, if we forget about the times of the year that it is occurring, and forget about the rate, we get just down to (1 + 1/n) to the n.1302

So, we can see what happens as n goes out to infinity--what number does this become?1310

It does stabilize to a number, as you can see from this graph here.1314

So, by the time it has gotten to 40, it starts to look pretty stable; it has this asymptote that it is approaching, so it is starting to become pretty stable.1317

We can look at some numbers as we plug in various values of n.1325

At 1, we get 2; at 10, we have 2.594; at 100, we have 2.705; at 1000, 2.717; at 10000, 2.718; at 100000, it is still at 2.718.1328

And there are other decimals there; but we see that it ends up stabilizing.1342

As we put more and more decimal digits, as n becomes larger and larger and larger, we see more and more decimal digits that e is going towards.1347

e is stabilizing to a single value, and we see more and more of its digits, every time we keep going with this decimal expansion.1356

So, as we continue this pattern, e stabilizes to a single number.1364

Now, it doesn't stabilize to a single number where we have finished figuring it; we keep finding new decimals.1367

But we see that decimals we have found so far aren't going to change.1372

e is 2.718281828...and that decimal expansion will keep going forever.1376

Just like π, the number e is an irrational number; its decimal expansion continues forever, never repeating.1382

So, that decimal expansion just keeps going forever, just like π isn't 3.14 (it is just keeps going forever and ever and ever).1389

So, e is the same thing, where we can find many of the decimals, but we can't find all of the decimals, because it goes on infinitely long.1399

Now, also, just like π, the number e is deeply connected to some fundamental things in math and the nature of the universe.1406

e is connected to the very fabric of the way that the universe, and just things, work.1414

So, π is fundamentally connected to how circles work; circles show up a lot in nature, in the universe.1418

π is connected to circles, and e is connected to things that are continuously growing--1423

things that are always growing, that don't take this break between growth spurts, but that are just always, always, always growing.1430

e gives us things that are doing this continual growth; e has this deep connection;1438

and if you continue on in math, you will see e a lot (and also if you continue on in science).1443

One application of e is to see how an account would grow if it was being compounded every single instant.1448

That idea, that we are not just doing it every year; not just every day; not just every minute; not just every second;1454

but every single instant--that gives us P (our principal amount) times ert.1459

The amount in our account is P times ert; we can also just remember this as "Pert"; Pert is the mnemonic for remembering this.1466

P is the principal, or we can just think of it as the starting amount--however much we started with.1474

r is the annual rate of interest, and it can even be used for things that aren't just annual rates,1478

but r is the annual rate of interest; and remember: we give that as a decimal.1483

If we give it as a percent, things will not end up working out.1487

And t is the number of years elapsed.1489

Now, this above equation, this one right here--this "Pert" thing--this can be used for a wide variety of things1493

that grow or decay continuously--things that are constantly growing or constantly decaying.1499

You will see it show up a lot in math and science as you go further and further into it.1503

It is very, very important--this idea of some principal amount, times e to the rate times the amount of time elapsed.1508

You can use it for a lot of things; and while we will end up, in these next few examples, using some other things1516

than just ert (with the exception of the examples that involve continuously compounded interest),1521

you can actually bend a lot of stuff that you have in exponents into using e.1525

So, it is easiest to end up just remembering this one, and then changing how you base your r around it.1530

Now, don't get too confused about that right now; we will see it more as we get into other things and logarithms,1535

and also just as you get further and further into math.1540

You will see how Pert is a really fundamental thing that gives us all of the stuff that is doing the growth.1542

Finally, exponential decay: so far, we have only seen exponential functions that grow as we go forward--1550

f(x) = ax, where our base, a, is greater than 1; so it gets bigger and bigger as we march forward.1556

But we can also see decay, if we look at 0 < a < 1--1562

if a is between 0 and 1--it is a fraction--it is smaller than 1.1566

Here are some examples: if we have 4/5x, we see that one in red;1571

1/2x--we see that one in blue; 1/10x--we see that one in green.1576

Notice how quickly the functions become very small as they repeatedly lose value because of the fraction compounding on them.1583

1/10 becomes very small by the time it has gotten to just 2; we have 1/102, which is equal to 1/100.1590

So, it becomes very, very small: by the time we are at (1/10)10, we are absolutely tiny.1598

Once again, it looks like it touches the x-axis, but that is just because it is a picture.1604

It never actually quite gets there; there is always a thin sliver of numbers between it.1608

But it gets very, very, very close; they will all become very, very small as the fraction on fraction on fraction compounds over and over.1611

Bits get eaten away each time the fraction hits, so it gets smaller and smaller and smaller.1621

But notice: if we go the other direction, we end up getting very large, just like normal exponential functions that grow, where a was greater than 1.1626

They got small when they went negative; they grew when they went positive,1635

because when they went negative, they flipped; we have that same idea of flipping.1639

If we have 1/10 to the -2, well, that is going to be 10/1 squared, which is equal to 100.1642

And that is why we see it blow up so quickly--it becomes very, very large, because we go negative for decay things.1651

But we will normally be looking at it as we go forward in time, which is why we talk about decay,1657

and things that are greater than 1 being growth, because we are normally looking at it1661

as we go forward--as we go to the right on our horizontal axis.1664

All right, let's look at some examples.1668

A bank account is opened with a principal of $5000; the account has an interest rate of 4.5%, compounded semiannually (which is twice a year).1670

How much money is in the account after 20 years?1678

So, what do we need? We go back and figure out the function we are using.1680

The formula is the one for interest compounded; so it is our principal, times 1 plus the rate, but divided by the number of times it occurs,1685

and then also raised to the number of times it occurs in the year, times the number of years that pass.1696

So, what are the numbers we are dealing with here?1701

We have a principal of $5000; we have a percentage rate of 4.5%, but we need that in decimal, so we have 0.045.1703

And what is the amount of time? The amount of time is 20 years.1717

If we do this with it going semiannually, twice a year, when we look at that, it will be n = 2.1722

a at 20 =...what is our principal? $5000, times 1 + the rate, 0.045 divided by the number of times it occurs in the year;1730

it occurred twice; n = 2, so divide by 2; raise it to the 2, times how many years? 20 years.1743

We go through that with a calculator; it comes out to 12175 dollars and 94 cents.1751

Now, what if we wanted to compound more often--what if it had been compounded quarterly or monthly or daily or continuously?1763

If it was compounded quarterly, it would occur four times in the year--every quarter of the year, every season--so n = 4.1770

So, we have 5000 times 1 + 0.045/4; that will be 4 times 20; we use a calculator to figure this out.1779

It comes out to 12236 dollars and 37 cents.1792

So, notice that we end up making a reasonable amount more than we did when it was compounded just twice in the year.1800

We are making about 50 dollars more--a little bit more than 50 dollars.1805

What if we have it do it monthly? How many months are there in a year?1810

There are 12 months in a year, so that would be n = 12.1813

5000 is our initial principal, times 1 + our rate, over 12 (I am losing room)...12 to the t...12 times t; so what is our t?1817

Our t was 20; sorry about that...12 times 20.1829

That will come out to be 12277 dollars and 33 cents.1836

What if we have it at daily--how many days are there in the year?1845

There are 365 days in a year, so that will be an n of 365.1849

So, at 365, we have 5000 times 1 + 0.045/365 (the number of times it occurs--365--the number of times it occurs in the year);1855

we had 20 years total; we simplify that out; we get 12297 dollars and 33 cents.1871

And what if we managed to do it every single instant--we actually had it compounding continuously?1881

Well, if n is equal to infinity, we are no longer using this formula here.1886

We change away from this formula, and we switch to the Pert formula, because that is what we do for compounded continuously.1890

That is going to be 5000 times e; what is our rate? 0.045; how many years? 20 years.1898

Once again, we punch that into a calculator: there will be an e key on the calculator--1906

you don't have to worry about memorizing that number that we saw earlier, because there is always an e key.1909

5000 times...oops, let's just get a number here; we are not going to end up doing this number, because it would be hard to do.1915

We will use a calculator; so let's just hop right to our answer.1926

We get 12298 dollars and 2 cents.1928

Finally, I would like to point out: notice that we ended up seeing reasonable amounts of growth1935

when we jumped from going only semiannually (twice a year) to four times a year.1940

And we also saw an appreciable amount of increase when we went from four times a year to twelve times a year-- when we went to monthly.1945

We got a jump of a little over 40 dollars.1953

When we managed to make it up to daily, we got a jump of about 20 dollars.1956

But going from daily to every single instant forever only got us a dollar.1960

So, we get better returns the more often it happens; but they end up eventually coming to an asymptote.1964

It increases asymptotically to this eventually stabilizes at a single value.1970

So, you won't see much difference between an account that compounds every single day and an account that compounds every single instant.1976

There won't be a whole lot of difference.1984

It is much better to have daily versus yearly, but daily versus continuously is not really that noticeable.1986

The second example: The day a child is born, a trust fund is opened.1992

The fund has an interest rate of 6% and is compounded continuously.1996

It is opened with a principal of $14000; what is the fund worth on the child's eighteenth birthday?1999

What formula will we be using? We will be using Pert.2004

The amount that we have in the end is equal to the principal that we started with, times e to the rate that we are at times t.2007

What is our principal? Our principal is 14000 dollars. What is our rate? Our rate was 6%.2014

We can't just use it as a 6; we have to change it to a decimal form, because 6 percent says to divide by 100; so we get 0.06.2023

Finally, what is the amount of time that we have?2030

In our first one, we are looking at a time of the eighteenth birthday--so 18 years; t = 18.2032

A principal of $14000 times e to our rate, 0.06, times the amount of years, 18 years--2039

we plug that into a calculator, and we see that, on his eighteenth birthday, the child has managed to get 41225 dollars and 51 cents.2050

So, that is pretty good; but what if the child managed to not need the money--didn't really want the money--2062

wanted to save it and maybe use it to buy a house when he was 30 (or put down a good down payment on a house when he was 30)?2067

At that point, if he was 30 before he took out the money, the child would have 14000;2073

it is the same setup, but we are going to have a different number of years--times 30.2079

That would end up coming out to 84000; it has more than doubled since he was 18--pretty good.2083

So, it has more than doubled; he has managed to make $84000 there.2092

That is not bad--he could get a good down payment on a house with that, so it is pretty useful.2097

But if he really didn't need the money--if he managed to not spend that money,2102

and he said, "I will use it as a retirement fund; that way I won't have to invest for my retirement at all--I already have it set up."2105

How much would he end up having at the age of 65?2110

We have 14000--the same setup as before--times e to our rate, 0.06, times our new number of years we are doing--it is 65 years.2114

And you would manage to have a huge 691634 dollars and 29 cents.2124

So, this points out just how powerful compound interest was.2134

We managed to start at 14000 dollars; but if we can avoid touching that money,2137

if we can just leave it for a very long time, we can get to very large values as the interest compounds on itself over and over again.2142

In 65 years, which is a very long time, we managed to grow from 14000 dollars to 691634 dollars--a lot of money.2148

And this gives us an appreciation for how important it is to make investments for retirement at an early age.2162

It is difficult when you are young; but if you manage to invest when you are young--2168

if you can wait on spending that money now--it can grow to very large amounts by the time you want to spend it to retire.2172

So, that is the benefit of investing early--being able to do that.2177

Also, it shows just how great, how useful, an interest rate is.2181

If that 6% was bumped up to 8% or 10%, we would see massive increases.2185

You can get a lot of increase if you can just get that percentage rate up another point or two--it is pretty impressive.2189

All right, the third example: The population of yeast cells doubles every 14 hours.2194

If the population starts with 100 cells, how many cells will there be left in two weeks?2199

So, this isn't compound interest, and it isn't continual growth, like we had before.2205

We might want to build our own here.2210

The population is doubling, so let's say n is the number of cells after some time.2212

We will set it up as a function--that makes sense; we are in "Exponential Function Land" right now.2217

So, n(t) is equal to...well, how many cells did we start with?2221

We started out with 100 cells, and we were told that it doubles.2226

So, we are going to have some "times 2," because we multiply it by 2.2231

How often does it do that? It does it every 14 hours.2234

So, if we have our number of hours, t = number of hours, t divided by 14 will be how many times it has managed to double.2237

After 14 hours, we have multiplied by 2 once.2249

After 28 hours, we have multiplied by 2 twice; we have 2 times 2 at 28 hours.2252

So, let's do a quick check and make sure that this is working out so far.2258

So, if we had n at 14 hours, we would have 100 times 214/14, which would simplify to 100 times 21.2262

So, we would get 200, so that part checks out.2273

Let's try one more, just to be sure: n(28)...if we had double double, then we know that we should be at 400, so we can see what is coming there.2276

So, 2 times 28/14...that simplifies to 100 times 22, which is equal to 100 times 4, or 400.2285

So, that checks out, as well; it passes muster--this makes sense as a way of looking at things.2295

So, as long as we have the amount of time we spent and the number of hours,2300

then we can see how many cells we have after that number of hours.2304

Now, we were told to figure out how many there will be in 2 weeks.2308

And we can assume that none of the cells die off, so the number just keeps increasing.2313

It is a question of how many times the population has gotten to double.2316

If that is the case, what number are we plugging in--it is n of how many hours?2320

Is it 2? No, no, it is not 2! Well, how many weeks...oh, 14 days? No, it is not 14.2324

What were we setting this up in? t was set in number of hours.2330

So, the question is how many hours we have on hand.2334

Let's first see how many hours 2 weeks is: how many days is that?2337

Well, that is going to be 2 many days in a week?...7, so that is 2 times 7 days.2344

How many hours is that? 2 times 7 times 24, or 14 times 24 hours, which we could then figure out with a calculator, and get a number of hours.2349

But we can actually just leave it like that, which (we will see in just a few moments) is a useful thing to do,2361

because we notice that there is a divide by 14 coming up; maybe it would be useful to just leave it as 14 times 24--a little less work for us.2365

So, 14 times notice: 14 times 24 is the number of hours in 2 weeks.2373

That is why we are plugging that in, because once again, the function we built,2384

our n(t) function that we built, was based on hours going into it.2388

We can't use any other time format.2392

100 times 2 to the 14 times 24 (is the number for t), divided by 14; look at that--the 14's cancel out.2395

We can be a little bit lazier--that is nice.2406

100 times 2 to the 24: we plug that into a calculator, and we get a huge 1677721600 cells.2408

That is more than one and a half billion cells: ones, thousands, millions, billions.2425

So we are at 1.6 billion cells--actually, closer to 1.7 billion cells.2432

This gives us a sense of just how fast small populations are able to grow.2440

And that is how populations grow: they grow exponentially, because each cell splits in half.2443

So, if we have one cell split in half to 2, and then each of those splits in half to 4,2449

and each of those splits in half to 8, this is going to do this process of exponentiation.2453

We are doing this through doubling, so we are going to see very, very fast growth.2458

And we actually see this in the real world.2461

We could also write this, for ease, as...we have 1, 2, 3, 4, 5, 6, 7, 8, that is the same thing...2464

we could write it as approximately 1.67x109 cells, 2474

so that we can encapsulate that information without having to write all of those digits.2482

That is scientific notation for us; all right.2486

The fourth and final example: The radioactive isotope uranium-237 has a half-life of 6.75 days.2488

Now, what is half-life? We would have to go figure that out, but luckily, they gave it to us right here.2495

Half-life is the time that it takes for one-half of the material of our isotope to decay and break down--to go through a process of decay.2499

If you start with one kilogram of U-237, how much will have not decayed after a year?2508

So, we are saying that, after 6.75 days, we will have half of a kilogram.2515

We start with one kilogram, and we know that, after every 6.75 days, we will have lost half of our starting material.2522

So, we will go down from one kilogram to half of a kilogram that has not decayed.2529

So, let's see if we can figure out a way to turn this into another function.2533

The...let's make it amount...the amount of our isotope that has not decayed, based on time,2537

is equal much did we start with? We started with 1 kilogram; times...what happens every cycle?2545

1/2...we halve it every time we put it through a cycle; so how fast is a cycle?2551

The number of days--we will make t into the number of days, because we see that we are dealing with days, based on this here.2559

So, t divided by 6.75--let's do a really quick check.2567

We check, because we know that, after 6.75 days, we should have 1/2 of a kilogram.2572

So, let's check that by plugging it in: a(6.75) is going to be 1 times 1/2 raised to the 6.75 over 6.75,2580

which is the same thing as just 1/2 to the 1, which equals 1/2.2592

So sure enough, it checks out--it seems like the way we have set this up passes muster,2596

because it is going to divide by half every time the 6.75 days pass.2601

So, if we plugged in double 6.75, it would divide by half twice, because it would be 1/2 squared.2604

It seems to make sense; we have set it up well; and we can see that this also can be just written as 1/2 times...2610

let's just leave it as it is; it gives us a better idea of how this works in general, for half-life breakdowns.2616

So, now we are going to ask ourselves how long--what is the time that we are dealing with?2621

In our case, t is one year; what is one year in days (because we set up our units as days,2625

because that is what our half-life was given to us in)?2632

One year is 365 days; so at the end of that, we plug in 365 = 1 (the amount that we started with),2634

times...the half-life will occur every 6.75 days (and we are still having 365 days go in).2643

We plug that all into a calculator, and we get the amazingly tiny number of 5.273x10-17 kilograms--a really, really, really small number.2652

To appreciate how small that is, let's try to expand it a bit more.2669

1 kilogram is 1000 grams; so that means that a kilogram is 103 grams.2672

We could also write this as 5.273x10...if it is 1000 grams for a kilogram, then that means we are going to increase by 3 in our scientific exponent.2681

So, in the scientific notation, we are now at 5.237x10-14 grams of our material.2695

which, if we wanted to write this whole thing out...we would be able to write it as 0.00000 (five so far) 00000 (10 so far) 000 (13)...2701

and let's see why that is the case--we can stop writing there--because if we were to bring that 10-14 here,2720

(and remember, it is in grams, because we had grams here), that would count as 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,2728

because we can move the decimal places 14 times to the right by having 10-14.2740

And that is how that scientific notation there is working.2745

Or alternatively, we could also write this with kilograms as the incredibly tiny 0.00000000000000005273 kilograms.2749

And if we counted that one out as well, we would have 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17...2767

so we have that 5.273x10-17 kilograms there, as well.2779

So, it is much easier to write it with scientific notation; that is also probably what a calculator would put out,2783

because it is hard to write a number like this, this long, on a calculator.2788

So, we are much more likely to see it in scientific notation, 5.273x10-17 kilograms,2792

which is an absolutely miniscule amount of radioactive material left, considering that we started at 1 kilogram.2799

That shows us how decay works.2805

All right, cool: we have a pretty good base in exponential functions.2807

Next, we will see logarithms, and see how logarithms allow us to flip this idea of exponentiation.2810

And then, in a little while, we will see how logarithms and exponential we can oppose the two against each other.2814

It is pretty cool--we can find out a lot of stuff with this.2819

All right, we will see you at later--goodbye!2821