WEBVTT mathematics/algebra-1/smith
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Welcome back to www.educator.com.
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In this lesson we are going to take a look at operations on numbers.
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When you hear me use that word operations, I'm talking about ways that we can combine together numbers or do something to a number.
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It gets very familiar things such as adding, subtracting, multiplying and dividing.
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As well as exponents and square roots.
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To combine numbers together, we use a lot of familiar operations in order to do so.
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Again, adding, subtracting, stuff like that.
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Note that depending on the types of numbers being used, certain rules applies that will help us put them together.
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The rules that I will focus on are the ones that involve positive and negative numbers.
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How should we deal with those negative signs? Let us see what we can do with addition.
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When you add numbers that have the same sign then you are looking at adding their absolute values together.
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The results if they do have the same sign or have the same sign as the original numbers.
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In other words, if you are adding the other two positive numbers its result will be positive.
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For adding together two negative numbers, then your result will be negative.
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Now if you happen to have two numbers that you are adding together and they are different in sign,
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then we will actually handle these using subtraction.
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Wait for the rules on what to do with positive and negative on my next slide.
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When dealing with subtraction and what we want to do is subtract the number that is smaller in absolute value
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from the number that is larger in absolute value and looking at each of their absolute values taking the smaller one from the larger one.
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When looking at the final result, the answer will have the same sign as whichever number was larger in absolute value.
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If your larger number was or the larger number in absolute value is negative, then the result will be negative.
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Remember that we will also think of adding negative numbers using subtraction in this way.
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In different way we are writing it but instead we are really using subtraction.
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The other two familiar operations that we have are multiplication and division.
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The way we handle a lot of signs with these ones are by thinking of this.
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If we multiply a negative × a negative then the result will be positive.
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A negative times a negative results positive.
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If we take two things that are different in sign such as negative times a positive, then the result will be negative.
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Now these are the same two rules that we end up using for division.
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If we divide a negative by a negative, you will get a positive.
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Negative divided by negative result positive.
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If they are different in sign such as negative divided by positive, then the result is negative.
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Keep these in mind when working with that your signs, multiplication and division.
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Let us talk about exponents.
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When we have an exponent, we can think of it as taking a number and multiplying it by itself in a certain number of times.
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For example, maybe I have 3⁷, 7 would be our exponent.
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I can interpret that as multiplying that number by itself 7 times or some other key vocabulary you want to pick up on.
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The exponent is that number raised next of the 3, exponent.
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In the base of the number is the number that we are actually being repeatedly multiplied, base.
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Once we see what number that is and how many times we need to multiply it, usually we can go ahead and simplify from there.
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There are some common exponents that we usually give other names.
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If I'm taking 5 and I'm raising it to the power of 2, 5 × 5, this is often said 5².
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Another good common one would be something like 2³, 2 × 2 × 2 that would be 8.
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I could also say that this is 2³.
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Key on these special words for some of these other powers.
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Another operation that we can do with numbers is taking the root of a number.
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The principal square root of a number is the non negative number (n)
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that you know if I were to say multiply it by itself, I would end up giving that (n).
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This seems a little funny special worry but let us see if I can describe it using (n) as an example.
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Let us say I wanted to find out the square root of 25.
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What I'm looking for is what number would multiply it by itself in order to get a 25? That has to be 5.
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It is what I'm talking about here, positive number such that when it is multiplied by itself you get that number underneath the root.
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For this reason we have a little bit of a problem with our negatives underneath the root.
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After all, what number would you multiply it by itself in order to get -16?
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We learned from our rules of positive and negative numbers that -4 and +4 would work to get 16
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but unfortunately they are different in sign and we need them to be exactly the same, that is not going to work.
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We are dealing with the principal square root because we are only interested in the positive numbers
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that when multiplied by themselves would give us that number.
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A perfect square is a number that is the square of a whole number and this one is usually reduced very nicely.
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For example the square root of 9 would be the example of a perfect square, as it reduces down to a nice whole number, 3.
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If you have a (a, b) being non negative real numbers,
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then there are a few different ways that you can say combine or rip apart those roots.
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3 multiplied underneath the same root and you can apply the root to each of those pieces.
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If you are dividing and you have a root then you can put it over each of its pieces in the numerator and in the denominator.
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You can use these rules in two different ways to simplify or combine words together.
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We will see that a lot in some future lessons.
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I'm pointing this out now so you do not make a common mistake.
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Do not try and split up your roots over addition or subtraction.
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We do not have a rule to do that yet or a good way to handle it.
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In fact, in this example I have written below you can see that the two are not equal by simply evaluating each side.
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9 + 16 would be a 25 and the square root of 25 is 5.
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Looking at the right side square root of 9 is 3, square root of 16 is 4.
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I'm putting those together I will get 7 and you can see that these things are not the same.
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Be very careful when working with your roots.
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Now that we know a few things about combining these, let us go through some examples and just practice with them.
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The first one I want to add together a -3 and a -6.
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I'm adding together two numbers that have exactly the same sign.
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I will look at their absolute value and add those together.
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The absolute value of 3 is 3, the absolute value of -6 is 6, if I combine those together I will get 9.
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Since I'm adding together numbers that have exactly the same sign, the result will also have the same sign.
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Adding two negative numbers my result is negative, -3+ -6 is a -9.
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Moving on, 19 + a -12, I want to think of this as a subtraction problem since their signs are different.
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How do I handle subtraction? Again I will look at their absolute value.
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The absolute value of 19 and the absolute value of a-12, 19, 12.
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I will subtract the smaller number from the larger number 19-12, what will that give me? I will get7.
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I have to determine what sign this should be.
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In subtraction we take the same sign as the larger number.
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19 was larger in absolute value, it was positive, I know my result is 7.
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If we do get a positive result as our answer, we do not write that positive sign up there.
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We are just having 7, we will assume that that is a positive 7.
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-8-11, an interesting way we can look at the problem, we could look at this as adding a -11.
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The reason why I have looked at it in that way is that I could use my rules for addition.
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If I'm adding things that have the same sign, I have looked at the absolute value of each of them, 8, 11.
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Then I can simply add up those two values and get 19.
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Since I'm adding two negative numbers, I know my result will also be negative, -19.
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Let us try another one, 8 - -13.
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When you subtract the negative this is another good situation that you could end up rewriting in a much simpler form.
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When you subtract the negative, you can change it into addition.
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This is 8 + 13 and now I'm adding together two positive numbers.
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You made a positive 21 and one more, negative the absolute value of a -4 + 9.
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Let us start inside those absolute values and see what we can do.
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I'm adding together two things but they have different signs.
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Let us look at the absolute value of a -4 and the absolute value of 9, 4, 9.
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We want to subtract the smaller value from the larger value, 9-4 and that result would be 5.
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I know what sign should that have, or the number that is larger in absolute value is 9.
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And that was a positive value over here, I'm looking at a 5.
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There are a lot of other things I have left out here so far,
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those would be the absolute values and that leading negative sign.
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Let us go ahead and put those in there now.
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I want to do the absolute value of 5, all that is simply 5.
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I still have that negative sign hanging out front and it is been there since the very beginning.
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I can see that after done evaluating this one all the way, my answer is actually a 5.
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Let us work on adding, multiplying and dividing the following numbers.
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We only have one rule to take care of that is when we are multiplying together two negative numbers, we get a positive.
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If we are multiplying together two numbers that are different in sign, the result should be negative.
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4 × -7, I just want to think of 4 × 7 that would be 28.
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Since they are different in sign, I know that this will be a -28.
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Moving on, -6 × -5, 6 × 5 would give me 30 and now here I have a - × - I know the result will be positive.
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But again we usually do not write that positive sign in there so just leave this as 30.
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-12 ÷ 3, 12 ÷ 3 would be a 4, negative ÷ positive would be negative, our result is a -4.
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One last one, -60 ÷ a -5, 60 ÷ 5 goes in there 12 times and negative ÷ negative is positive, my result is a 12.
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You need to be very careful with these rules for multiplication, make sure you have these memorized.
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Let us do a few involving our exponents.
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Remember we can think of these as repeated multiplication.
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2/3⁴, it can be the same as 2/3 × 2/3 × 2/3 × 2/3, we are doing it 4 times.
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I simply multiply it across the top and across the bottom 2 × 2 × 2 × 2 would be 16.
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Then I have 3 × 3 × 3 × 3 = 81.
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All the numbers here are positive so I know my result is positive.
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Here is a tricky one, -5 ×-5 × -5, let us take this two other times so we can what is going on here.
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Here I have a -5 × -5 the result of taking 5 × 5 would be 25.
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Taking a negative × negative I would know that this would be 25.
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That looks good, let us go ahead and work in this last value of -5.
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We want to multiply 25 × -5, the result there would be 125.
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Since I'm multiplying a positive × a negative result is -125, -125 would be my answer.
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The next one looks very similar but it is actually very different.
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That one is 2³ and that negative sign is just out front of all that.
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We recognize that the 2 is the base and that the negative is not included in that base
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since there is no parenthesis given to group it in that way.
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We want to figure this one as 2 × 2 × 2, I have multiplied it out three times.
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And as for what to do with that negative sign, if we have not put it up front it is a long pretty ride.
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2 × 2 × 2 that would give me an 8, all of those numbers are positive, 8.
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Of course let us put our negative sign out front since it was out front at the very beginning.
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We can see that our result is -8.
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Now on to some square roots, these ones we simply just want to break them down and simplify them as much as possible.
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The first one I have the square root of 64.
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Think for yourself what number would you have to multiply it by itself in order to get 64?
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I have a couple of options that could be 8 and 8, that would have given me 64.
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Or it could be -8 × -8, that would also give me 64.
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We are only interested in the positive values that do so, let us not worry about those -8.
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We will say that our answer for the square of 64 is 8.
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Moving on, the square root of 169 ÷ 81.
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This one we can use one of our rules and break up the root over the top and over the bottom.
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Now we can end up taking the root of each of these individually.
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What number multiplied by itself would give us a 169? That have to be a 13.
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A number that would be multiply by itself to get an 81? 9.
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The answer in this one is 13/9.
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Continuing on, this one has a negative sign out front but that is not underneath the root.
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I would not worry about it just yet instead let us just focus on the square root of 36.
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That would be 6, of course we will go ahead and put our negative sign and see that our final result is a -6.
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One more, this last one involves the square root.
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What two numbers when multiplied together would give us a -49 remember they must be the same.
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We got a few problems, do not we? If I try and use 7 and 7 that would give me 49, that does not work.
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If I try and use a couple of-7, that does not work, that still gives me 49.
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I can not use one positive and one negative even though those give me a -49.
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Those are not the same sign, one is positive and one is negative.
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What is going on here, if you remember about your types of numbers, these are imaginary numbers.
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I will leave that one just as it is until we learn about simplifying imaginary numbers in some later lessons.
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Some various different ways that you can go ahead and combine numbers using some very familiar operations.
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Remember that most of the rules that I covered will give you some tips on what to do when they are different in signs.
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Positive, negative, negative, positive and all those will be handy in figuring out the overall sign of your answer.
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