## Discussion

## Study Guides

## Practice Questions

## Download Lecture Slides

## Table of Contents

## Transcription

## Related Books

### Properties of Exponents

- These properties are used throughout this course. Learn them well.
- For a monomial to be in simplified form, each base must occur only once, all fractions must be simplified, there can be no negative exponents, and there can be no powers of powers.

### Properties of Exponents

^{3}z

^{4})

^{2}(z

^{2}y

^{2}x)

^{4}

- Distribute the exponents outside the parenthesis, multiply the exponents.
- (x
^{2}y^{6}z^{8})(z^{8}y^{8}x^{4}) - Add the exponents since variables are multiplying each other

^{2 + 4}y

^{6 + 8}z

^{8 + 8}= x

^{6}y

^{14}z

^{16}

^{5}b

^{6})

^{2}(a

^{2}b

^{3})

^{ − 3}

- Distribute the exponents outside the parenthesis, multiply.
- (a
^{10}b^{12})(a^{ − 6}b^{ − 9}) - Add the exponents since variables are multiplying each other

^{10 + ( − 6)}b

^{12 + ( − 9)}= a

^{4}b

^{3}

^{3}b

^{2}c

^{2})

^{2}(a

^{2}b

^{3}c

^{3})

^{ − 4}

- Distribute the exponents outside the parenthesis, multiply.
- (a
^{6}b^{4}c^{4})(a^{ − 8}b^{ − 12}c^{ − 12}) - Add the exponents since variables are multiplying each other
- a
^{ − 2}b^{ − 8}c^{ − 8} - You cannot have negative exponents, therefore, bring the variables to the denominator and change the exponent from negative to positive

^{2}b

^{8}c

^{8})]

^{3}b

^{ − 2}c

^{2})

^{ − 2}(a

^{2}b

^{ − 3}c

^{3})

^{ − 4}

- Distribute the exponents outside the parenthesis, multiply.
- (a
^{ − 6}b^{4}c^{ − 4})(a^{ − 8}b^{12}c^{ − 12}) - Add the exponents since variables are multiplying each other
- a
^{ − 6 + ( − 8)}b^{4 + 12}c^{ − 4 + ( − 12)}= a^{ − 14}b^{16}c^{ − 16} - You cannot have negative exponents, therefore, bring the variables to the denominator and change the exponent from negative to positive

^{16})/(a

^{14}c

^{16})]

^{4}y

^{3})/(x

^{2}y

^{4})] )

^{ − 3}

- Distribute the exponent into the parenthesis. Multiply the exponents.
- ( [(x
^{ − 12}y^{ − 9})/(x^{ − 6}y^{ − 12})] ) - To get rid of negative exponents, switch from numerator to denominator and denominator to numerator. Change the sign of the exponent.
- Alternatively, you may use the definition [(x
^{n})/(x^{m})] = x^{n − m} - ( [(x
^{6}y^{12})/(x^{12}y^{9})] ) - Now that we have positive exponents, cancel out 6 x's from the numerator and denominator, as well as 9 y's from denominator and numerator.

^{3})/(x

^{6})]

^{2}y

^{4})/(x

^{4}y

^{3})] )

^{ − 3}

- Distribute the exponent into the parenthesis. Multiply the exponents.
- ( [(x
^{ − 6}y^{ − 12})/(x^{ − 12}y^{ − 9})] ) - Use the definition [(x
^{n})/(x^{m})] = x^{n − m} - x
^{ − 6 − ( − 12)}y^{ − 12 − ( − 9)} - A negative times a negative equals a positive
- x
^{ − 6 + 12}y^{ − 12 + 9} - Simplify
- x
^{3}y^{ − 3} - Cannot have negative exponents, therefore, bring the y to the denominator.

^{3})/(y

^{3})]

^{ − 2}y

^{4}z)/(x

^{4}y

^{ − 3}z

^{ − 3})] )

^{ − 3}

- Distribute the exponent into the parenthesis. Multiply the exponents.
- ( [(x
^{6}y^{ − 12}z^{ − 3})/(x^{ − 12}y^{9}z^{9})] ) - Use the definition [(x
^{n})/(x^{m})] = x^{n − m} - x
^{6 − ( − 12)}y^{ − 12 − (9)}z^{ − 3 − (9)} - A negative times a negative equals a positive.
- x
^{6 + 12}y^{ − 21}z^{ − 12} - Simplify
- x
^{18}y^{ − 21}z^{ − 12} - Cannot have negative exponents, therefore, bring the y and z to the denominator.

^{18})/(y

^{21}z

^{12})]

^{ − 2}y

^{ − 4}z

^{ − 4})/(x

^{ − 4}y

^{ − 3}z

^{ − 2})] )

^{ − 3}

- Distribute the exponent into the parenthesis. Multiply the exponents.
- ( [(x
^{6}y^{12}z^{12})/(x^{12}y^{9}z^{6})] ) - Use the definition [(x
^{n})/(x^{m})] = x^{n − m} - x
^{6 − (12)}y^{12 − (9)}z^{12 − (6)} - Simplify
- x
^{ − 6}y^{3}z^{6} - Cannot have negative exponents, therefore, bring the x to the denominator.

^{3}z

^{6})/(x

^{6})]

^{12}y

^{13}z

^{14})/(x

^{10}y

^{ − 3}z

^{ − 20})] )

^{0}*3x

^{ − 5}

- Recall that anything raised to the zero power is always 1.
- 1*3x
^{ − 5}= 3x^{ − 5} - Cannot have negative exponents, therefore, bring the x to the denominator.

^{5})]

^{ − 10}y

^{0}z

^{ − 12})/(x

^{ − 12}y

^{5}z

^{ − 10})]

- Anything raised to the zero power is always 1. Therefore, eliminate y
^{0}. - [(x
^{ − 10})/(z^{ − 12})] - x
^{ − 12}y^{5}z^{ − 10} - Use [(x
^{n})/(x^{m})] = x^{n − m} - [(x
^{ − 10 − ( − 12)}z^{ − 12 − ( − 10)})/(y^{5})] - Simplify
- [(x
^{ − 10 + 12}z^{ − 12 + 10})/(y^{5})] - = [(x
^{2}z^{ − 2})/(y^{5})]

^{2})/(y

^{5}z

^{2})]

*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.

Answer

### Properties of Exponents

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
- Simplifying Exponential Expressions
- Negative Exponents
- Properties of Exponents
- Negative Exponents
- Mutliplying Same Base
- Dividing Same Base
- Raising Power to a Power
- Parentheses (Multiplying)
- Parentheses (Dividing)
- Raising to 0th Power
- Example 1: Simplify Exponents
- Example 2: Simplify Exponents
- Example 3: Simplify Exponents
- Example 4: Simplify Exponents

- Intro 0:00
- Simplifying Exponential Expressions 0:09
- Monomial Simplest Form
- Negative Exponents 1:07
- Examples: Simple
- Properties of Exponents 3:06
- Negative Exponents
- Mutliplying Same Base
- Dividing Same Base
- Raising Power to a Power
- Parentheses (Multiplying)
- Parentheses (Dividing)
- Raising to 0th Power
- Example 1: Simplify Exponents 7:59
- Example 2: Simplify Exponents 10:41
- Example 3: Simplify Exponents 14:11
- Example 4: Simplify Exponents 18:04

### Algebra 2

### Transcription: Properties of Exponents

*Welcome to Educator.com.*0000

*Today, we are going to start talking about polynomial functions; and we are going to begin with some review of properties of exponents.*0002

*So, recall that, when you need to simplify an exponential expression, you need to write the expression without parentheses or negative exponents.*0009

*Also recall that the monomial is in simplest form if each base appears only once,*0020

*if there are no powers of powers, and all fractions are simplified.*0041

*And recall that, in a polynomial, the terms are monomials.*0055

*OK, first reviewing the concept of negative exponents: if a does not equal 0*0067

*(and we have that limitation because we don't want to have a 0 in the denominator),*0073

*a to a negative power equals 1 over a to that power.*0077

*So, a ^{-n} equals 1/a^{n}: for example, if I had y^{-4}, I could rewrite that as 1/y^{4}.*0082

*Now, let's look and think about why this would be.*0103

*If you have something like x ^{3}/x^{5}, that would actually give you x times x times x, all over x times x, and on 5 times.*0107

*So, if I go ahead and look at what that would be, I could cancel out these x's (the first three).*0122

*And I would end up with 1/x times x, or 1/x ^{2}.*0130

*I could look at this another way: I could use my rules for dividing exponents of the same base.*0137

*And those rules would tell me that what I need to do is take x ^{3 - 5}; so I need to subtract the exponents.*0142

*And this is going to give me x ^{-2}; well, since these two are equal, these two must be equal.*0154

*Therefore, 1/x ^{2} equals x^{-2}; and that is why we say that these two are equivalent.*0161

*a ^{-n} equals 1/a^{n}.*0169

*And remember: in order to simplify exponential expressions, you need to make sure that there are no negative exponents.*0172

*And using this rule is how we get rid of those negative exponents.*0179

*OK, reviewing properties of exponents: each of these is covered in detail in Algebra I--this is just a brief review.*0186

*The first property is the one that we just talked about, which is negative exponents: a ^{-n} = 1/a^{n}.*0194

*The second property is a review of multiplying, where you have exponents with the same base.*0204

*So, if you have exponents with the same base, such as x ^{3}, and you are asked*0211

*to multiply that times something like x ^{4}, you accomplish that by simply adding the exponents to get x^{7}.*0216

*Division: to divide exponential expressions with the same base (I have two monomials here with the same base),*0225

*what I am going to do is subtract the exponents, again with the limitation*0233

*that a does not equal 0, because we cannot have 0 in the denominator, since that would be undefined.*0237

*So, if I had something such as y ^{4}/y^{6}, this is going to give me y^{4 - 6}, or y^{-2}.*0244

*And today, again, I am reviewing these; but we are also going to go on and apply the properties more to negative exponents than we did in Algebra I.*0256

*And you could further simplify this by writing it as 1/y ^{2}.*0268

*Now, raising a power to a power: when you raise a power to a power, you are going to do that by multiplying the exponents.*0272

*So, if I have x ^{4} raised to the second power, I am going to rewrite this as x^{4 x 2}, or x^{8}.*0280

*And again, this works with negative exponents as well.*0292

*If I have z ^{3}, raised to the -2 power, it is going to give me z^{3 x -2}, or z^{-6}.*0296

*So, this works for negative exponents.*0308

*Parentheses: recall that, in order to simplify exponential expressions, we need to get rid of parentheses.*0312

*So, if you have something such as ab, and that whole expression is raised to the m power, you can rewrite this as a ^{m} times b^{m}.*0318

*So, if I have something such as 3x, and it is squared, I can rewrite this as 3 ^{2}x^{2}, which would give me 9x^{2}.*0331

*Again, parentheses, but now talking about division: if I have a divided by b,*0348

*all raised to the n power, I can rewrite that as a ^{n} divided by b^{n},*0354

*with the restriction that b, since it is in the denominator, cannot equal 0.*0358

*For example, if I have x/y raised to the fourth power, I could rewrite that as x ^{4} divided by y^{4}.*0362

*Finally, something raised to a zero power; if I have a raised to a 0 power, it equals 1.*0376

*However, we again have the restriction where a cannot equal 0, because this is not defined.*0384

*So, we don't work with that.*0391

*Now, let's think a little bit more deeply about why this is true.*0394

*We are saying that a ^{0} equals 1; why is that so?*0397

*Well, it is simply because we defined it that way, to make everything work out, and all the rules be consistent.*0401

*And you can look at it this way: let's say I have y ^{3} over y^{3}.*0408

*Well, using my rule of division that I just talked about, when I have exponents with like bases,*0414

*I could say, "OK, this is y ^{3 - 3}, or y^{0}."*0421

*Well, I could also look at it another way: y ^{3}/y^{3} equals y times y times y, all over y times y times y.*0427

*OK, so now these cancel out; and what this gives me is 1/1, or 1.*0438

*Now, to make everything work out and have everything (all these rules) be consistent, what I am saying*0446

*is that these two are equal; and I used a rule here and a rule here, and I got these two different things;*0452

*therefore, these two things must be equal, in order to be able to use all of these rules.*0459

*So, we say that any number to the 0 power is 1.*0467

*Now, applying these rules to simplify some exponential expressions:*0480

*recall that, in order to be in simplest form, I need to make sure there are no negative exponents (which there are not),*0484

*no parentheses (I have parentheses), and no powers raised to powers (I have that, so I need to take care of all that).*0490

*So, I am going to start out by recalling that a power raised to a power is equal to a ^{m x n}.*0497

*So, let's first get rid of these parentheses by saying that I actually have*0510

*(a ^{2})^{4} times (b^{3})^{4} times (c^{2})^{4}*0515

*times (b ^{2})^{3} times (c^{3})^{3} times a^{3}.*0527

*Now, all I need to do is multiply these out; and this is going to give me...I have powers raised to powers,*0537

*so a ^{8} times b^{12} times c^{8} times...2 times 3 is 6, so that is b^{6},*0547

*times c ^{9}, times a^{3}; so, this monomial is still not in simplest form,*0558

*because I still have some bases that are duplicate here.*0566

*Each base should be represented only once in the monomial.*0570

*So, in order to simplify this, I need to multiply.*0573

*And what I can do is recall my rule for multiplication when exponents have the same base.*0577

*a ^{m} times a^{n} equals a^{m + n}.*0584

*So, what I can do is say, "OK, I have a to the eighth power, and I have a to the third power, so I am going to add those exponents."*0590

*I have b to the twelfth power, and I have b to the sixth power; so I am going to add those.*0597

*I have c to the eighth, and I have c to the ninth; so I am going to add those.*0603

*And this is going to give me a ^{11}b^{18}c^{17}.*0608

*So, this is now in my simplest form; and I accomplished that by getting rid of the parentheses and using my rule for raising the power to a power.*0616

*So, I multiplied each of these times its power; and then I found that I had bases represented more than once.*0626

*So, I multiplied the expressions that had like bases by adding the exponents.*0633

*OK, simplify: here we have parentheses; we have powers to powers; and we have negative exponents.*0641

*So, I need to get rid of all that, first by applying the power to each term inside the parentheses.*0648

*This is going to give me 2 ^{8} times (x^{2})^{8} times (y^{3})^{8}*0654

*times (z ^{4})^{8} times x^{-2}y^{-2}(z^{2})^{-4}.*0664

*OK, simplifying: if you work this out to multiply 2 by itself 8 times,*0674

*you would find that you are going to get 256 for that one.*0681

*Now, here I have 2 times 8; that is 16; 3 times 8 is 24; z ^{4 x 8}--that is 32.*0687

*And then here, I have a bunch of negative exponents: x ^{-2}y^{-2}...z^{2 x -4} is z^{-8}.*0707

*Now, recall that, if I have a ^{-n}, this equals 1/a^{n}.*0716

*So, in order to eliminate these negative exponents, I am going to move all three of these into the denominator.*0722

*But this is actually an error right here; OK, let me correct that.*0745

*This is z squared to the negative 2, so this would be 2 times -2; that is actually negative 4; so this is z to the fourth power right here.*0748

*Now, I can further simplify by dividing; recall that dividing exponential expressions with like bases, you subtract the exponents.*0757

*I can do that here: this is going to give me 256x ^{16 - 2}, times y^{24 -2}, times z^{32 - 4}.*0768

*Now, I do my subtraction to get x raised to the fourteenth power, y to the twenty-second power, and z to the twenty-eighth.*0788

*And this is my expression in simplest form; and I can verify that, because each base is represented only once;*0799

*there are no powers to powers; and there are no negative exponents; and there are no parentheses.*0806

*Now, this was kind of complicated; but I started out by applying the exponents to each of the numbers and variables inside the parentheses.*0811

*And I multiplied using my rule for a power to power, which is (a ^{m})^{n} = a^{mn}.*0823

*OK, once I did that, I saw negative exponents; and I used this rule to move those into the denominator, so they became positive.*0834

*And then, I had like bases, so I used my rule for division, in order to further simplify, getting this as the final result.*0841

*OK, simplify: here I have a negative power, and I have items inside of parentheses raised to that negative power.*0851

*Recall this rule: if I have a/b, all raised to a certain power, this equals a to that power, over b to that power.*0860

*So, what I can do is apply this -4 to the numerator and to the denominator, separately.*0870

*So, this is going to give me a ^{3}b^{2}x^{4}, all to the negative fourth power,*0876

*over x ^{2}y^{3}c^{3}, all to the negative fourth.*0882

*OK, once I have that, I am going to use my power to a power rule: (a ^{m}) raised to the n power equals a^{mn}.*0889

*OK, so I am applying the -4 to each item inside the parentheses and doing the same thing in the denominator,*0899

*and then, multiplying the exponents to get a ^{-12}b...2 times -4 gives me -8;*0925

*and then here, I have 4 times -4 is going to give me -16: x ^{-16}; x^{-8}y^{-12}c^{-12}.*0936

*Before I go any farther and deal with the negative exponents,*0954

*I first notice that I have some simplifying I can do, because these two have the same base.*0957

*Therefore, I can divide; and dividing will require me to subtract the exponents (dividing with like bases).*0962

*a ^{-12}b^{-8}x^{-16 -}...and this is a negative, so it is - -8.*0971

*And now, in the bottom, I just have y ^{-12}c^{-12}.*0982

*OK, this is going to give me a ^{-12}b^{-8}...-16, and this is minus - -8, so it is really + 8;*0986

*-16 plus 8 is going to leave me with -8; in the denominator, I have y ^{-12}c^{-12}.*0997

*Now, I have a lot of these negative exponents.*1007

*Recall the rule for negative exponents: a ^{-n} = 1/a^{n}.*1010

*Therefore, if I move everything in the numerator to the denominator, those exponents will become positive.*1015

*For the ones in the denominator, I have to switch those to the numerator for them to become positive.*1022

*Therefore, I will end up with y ^{12}c^{12} over a^{12}b^{8}x^{8}.*1027

*OK, so this is simplest form, because I have no parentheses; I have no powers to powers; and I have no negative exponents.*1038

*I started out by splitting the numerator and the denominator by applying this power separately to each.*1046

*Then, I applied the power to each item inside the parentheses to come up with this.*1052

*Then, I noticed that I had the x's, and they had the same base, so I divided x ^{-16} by x^{-8} to get this.*1060

*And finally, I was eliminating these negative powers by using this rule*1070

*and switching the items in the numerator and the denominator, to get my simplest form.*1076

*OK, we are asked to simplify; and we have bases that are represented twice, and we also have negative exponents that we need to get rid of.*1083

*First, I am going to start out by dividing; I have a ^{-3}, and then that is going to be,*1094

*using my rule for division, a ^{m} divided by a^{n} equals a^{m - n}...*1101

*-3 minus -4; b ^{2 - -3}...you have to be careful with the signs with all these negatives...*1108

*and then c ^{-2 - -4}; let's take care of these signs.*1121

*This is -3; a negative and a negative actually gives me + 4.*1128

*And this is 2, minus -3, so that is + 3; and then c, minus -4, is + 4.*1133

*So, this gives me -3 + 4 (is just 1, so I leave that as a); b: 2 + 3 is 5; and then c: -2 + 4 is 2.*1143

*So, this started out looking very messy; but actually, once you divided, it took care of all of those negative exponents.*1153

*That concludes this lesson of Educator.com on exponential expressions.*1163

*And I will see you soon!*1168

3 answers

Last reply by: Dr Carleen Eaton

Sun Aug 12, 2018 9:35 PM

Post by John Stedge on June 4 at 04:37:32 PM

If a parenthesized group is being multiplied do you distribute to the parenthesized group or not.

1 answer

Last reply by: Dr Carleen Eaton

Sun Apr 21, 2013 9:21 PM

Post by emily vita on April 13, 2013

in ex. of third the power of z is -2 not -4