Lesson Video: Laws of Exponents | Nagwa Lesson Video: Laws of Exponents | Nagwa

# Lesson Video: Laws of Exponents Mathematics • First Year of Preparatory School

In this video, we will learn how to apply the laws of exponents to multiply and divide powers and work out a power raised to a power.

17:33

### Video Transcript

In this video, we will learn how to apply the laws of exponents to multiply and divide powers and also how to work out a power raised to a power. We will recap exponents and go through some questions, including a question that looks like this. Using the laws of exponents, weโll see how questions like this can be made so much more manageable. Letโs begin by recalling the basics of exponents or powers.

The small number here is often referred to as the index or exponent. Three raised to the fourth power or three to the power of four means that we have four threes all multiplied together. We could calculate the value of this in several ways. We could begin by multiplying three times three, which is nine. Multiplying nine by another three would give us 27. And then multiplying 27 by the final three would give us 81. Alternatively, we couldโve got our three times three, which is nine, and multiplied by the second lot of three times three, which is nine again. And nine multiplied by nine is 81. So letโs take a look at what happens when we multiply two values with exponents.

Fill in the blank: Negative two to the seventh power times negative two to the fifth power equals negative two to the power of what.

In this question, weโre asked to find the index or power that negative two in the answer would be raised to. We can start by thinking about what the value of negative two to the seventh power would look like. This would mean that we would have negative two seven times. And all of these would be multiplied together. In the same way, negative two to the fifth power means that we have five negative twos all multiplied together. And our negative two to the seventh power is multiplied by our negative two to the fifth power. So that means our seven negative twos are multiplied by our five negative twos.

So how many negative twos have we got in total now? Well, now, we have 12, which means that we could write this as negative two raised to the power of 12. And so our missing exponent value is 12. We could also have answered this using the exponent rule that ๐ฅ to the power of ๐ multiplied by ๐ฅ to the power of ๐ is equal to ๐ฅ to the power of ๐ plus ๐. The value of ๐ฅ in the original question would be negative two; our two exponents ๐ and ๐ are seven and five. So our exponent value at the end is equal to ๐ plus ๐ which would be seven plus five giving us 12, confirming our original answer.

In this question, we saw our first exponent rule that ๐ฅ to the power of ๐ multiplied by ๐ฅ to the power of ๐ is equal to ๐ฅ to the power of ๐ plus ๐. We know this to be true because if we write our first ๐ฅ to the power of ๐ as ๐ lots of ๐ฅ multiplied together and our second value of ๐ฅ to the power of ๐ as ๐ lots of ๐ฅ, then when we add these together, we know that we will have a total of ๐ plus ๐ lots of ๐ฅ. As these will all be multiplied together, we can write this as ๐ฅ to the power of ๐ plus ๐.

In the next question, weโll have a look at dividing values with exponents.

Is three to the 74th power over three to the 75th power greater than, less than, or equal to three?

Letโs begin this question by thinking about what three to the 74th power and three to the 75th power will actually involve. We can recall that, for example, three to the fourth power means that we write three down four times and multiply it together. So three to the 74th power means that there would be 74 lots of three multiplied together. Three to the 75th power would be 75 lots of three multiplied together. Actually, calculating the value of either of these would take a very long time. And in this question, we also need to divide. But letโs have a look at what happens if we cancel values from the numerator and denominator.

We could cancel off the first threes and the second two threes and so on until we have canceled off 74 threes on the numerator and the denominator, leaving us with just three on the denominator. We know that this will be equivalent to one-third. So three to the 74th power over three to the 75th power is equal to one-third. There is another way we couldโve calculated this, though. The quotient law of exponents tells us that ๐ฅ to the power of ๐ over ๐ฅ to the power of ๐ is equal to ๐ฅ to the power of ๐ minus ๐.

Here, the value of ๐ฅ would be three and our two exponents ๐ and ๐ would be 74 and 75. So our answer here would be three to the power of 74 minus 75. And thatโs equivalent to three to the power of negative one. Three to the power of negative one is the reciprocal of three, and thatโs the same as one-third. Our question asks if this is greater than, less than, or equal to three. One-third is less than three. And so thatโs our answer.

We can pause for a second and add this second quotient law to our laws of exponents. ๐ฅ to the power of ๐ over ๐ฅ to the power of ๐ equals ๐ฅ to the power of ๐ minus ๐. We can, of course, see the equivalent question written with the division sign instead, ๐ฅ to the power of ๐ divided by ๐ฅ to the power of ๐. The value of this would still be the same. So far, weโve seen what happens when we multiply values with exponents and divide values with exponents. But what happens when we take a power of a power? Letโs find out in the next question.

Find the value of negative three to the third power times three squared to the third power.

In this question, we can see that our values here are values that have exponents or powers. Itโs worth noting that the wording of โfind the valueโ means that weโre not looking for an answer that has an exponent. For example, if we found that this was three to the third power, then weโd need to give the answer as 27 instead. We can recall that negative three to the third power means that we have three lots of negative three multiplied together. When weโre looking at three squared to the third power, we can recall that three squared means three times three. And so writing this to the third power means that we have three times three three times.

This is all becoming rather complicated to say. So letโs compare this with the rule that we have for finding the power of a power. In this case, ๐ฅ to the power of ๐ to the power of ๐ is equal to ๐ฅ to the power of ๐๐. If we look at three squared to the third power, our value of three here is our ๐ฅ-value. So our answer will be three to the power of two times three, three to the power of six. And we did indeed write six lots of three and we multiplied together. So letโs work out our value when we have negative three to the third power multiplied by three squared to the third power or three to the power of six.

Letโs start with negative three times negative three. We know that two negative values multiplied together will give us a positive value, so weโll start with nine. We then multiply by the next negative three and nine times negative three is negative 27. Continuing to multiply by three, we have negative 27 times three, which is negative 81. We can continue to multiply by three bringing down our threes as we go until we get to a final answer.

Itโs worth noting that there are, of course, a number of ways in which we couldโve multiplied. We know that negative three times negative three times negative three is negative 27 and three times three times three is 27. And so multiplying negative 27 by 27 would give us negative 729, and multiplying that by the final 27 would also give us the value of negative 19683. And this will be our final answer for the value of negative three to the third power times three squared to the third power.

Letโs pause for a minute to update our notes. We saw that we can take the power of a power, such as ๐ฅ to the power of ๐ to the power of ๐, by multiplying the exponents to give us ๐ฅ to the power of ๐๐. So far in this video, weโve been looking at integers raised to a power. Now, letโs have a look at some fractions raised to a power.

Which of the following is equal to negative one and a half to the third power times negative one and a half squared? Option (A) seven and 19 over 32, option (B) negative seven and 19 over 32, option (C) 32 over 243, option (D) negative 3125 over 1024, or option (E) negative 243.

In this question, weโre asked to multiply these two values with exponents. When weโre working with fractions and writing these with exponents, itโs always a good idea to make sure that the fractions are written as top-heavy or improper fractions rather than a mixed number. In both these cases, we have negative one and a half. And so we can write this as negative three over two to the third power times negative three over two squared.

Itโs useful to remember the product law of exponents when weโre multiplying a value ๐ฅ to the power of ๐ by ๐ฅ to the power of ๐. Then we add the exponents to give us a value of ๐ฅ to the power of ๐ plus ๐. In this case, our ๐ฅ-value will be negative three over two and our exponent can be calculated by adding our indices three and two, giving us five. So we have negative three over two to the power of five. What weโve done then is simplified our calculation to give a value with an exponent. However, if we look at the answer options, what weโre looking for here is to actually find the value of negative three over two to the fifth power.

Breaking down what it actually means to have negative three over two to the fifth power, that means we have five lots of negative three over two all multiplied together. As a quick aside, when weโre working with a fraction like negative three over two, there are several ways in which we can write this, firstly, with the negative sign in front of the fraction or with the negative three on the numerator or with the negative on the bottom to make negative two. Although the final form is mathematically correct, we tend to either have the negative sign on the numerator or externally to the fraction.

Returning to the problem then, we could think of this as a giant fraction with five lots of negative three multiplied on the numerator and five lots of two multiplied on the denominator. So calculating the value then, negative three times negative three is nine. Another negative three times another negative three is nine. Nine times nine is 81. And 81 times negative three gives us negative 243 on the numerator. On the denominator, we know that two times two is four, four times four is 16, and multiplied by another two gives us 32 on the denominator.

We now need to write our improper fraction of negative 243 over 32 as a mixed number. When we perform some long division of 243 divided by 32, we get the value of seven with a remainder of 19. The value of seven represents the whole number part of our answer. The remainder forms the numerator of our fraction. And as weโve divided by 32, then that will be our denominator. We mustnโt forget that we were dividing negative 243 by 32. So our value will be negative seven and 19 over 32. We can see that this is the answer given in option (B).

Weโll now look at one final question involving exponents.

Calculate negative three and a fifth to the seventh power times negative one and a half to the sixth power over negative 16 over five to the sixth power times negative three over two to the fourth power, giving your answer in its simplest form.

Although this question can look quite difficult, weโre going to start by writing the mixed numbers as improper fractions and then see what exponent laws we could apply. Letโs start with the mixed number negative three and one-fifth. The whole number part of three is made up of three five-fifths plus one-fifth left over would give us sixteen fifths. We werenโt dealing with just three and a fifth; it was negative three and a fifth. So weโll have negative 16 over five to the seventh power.

Next, negative one and a half is equivalent to negative three over two and thatโs to the sixth power. We can keep the denominator as it was as our fractions are improper here and not mixed members. At this point, we might hopefully begin to notice something about what weโve written. We can see that we have a negative 16 over five on the numerator and denominator. And we also have a negative three over two on the numerator and denominator. At this point, we can start to think if we can cancel on the top and the bottom of this fraction.

Letโs consider the first part of this fraction. We can use the quotient exponent law ๐ฅ to the power of ๐ over ๐ฅ to the power of ๐ equals ๐ฅ to the power of ๐ minus ๐, because we have the same value, which is taken to a power. Thatโs our value of ๐ฅ. Our answer will have our ๐ฅ. Thatโs negative 16 over five raised to a power. To work out this power, we have seven take away six, which gives us one. Negative 16 over five to the power of one is the same as negative 16 over five.

Now, letโs simplify the second part of this fraction. Using the same rule of exponents to simplify negative three over two to the sixth power over negative three over two to the fourth power gives us negative three over two squared, remembering that the squared comes from six subtract four. So now, we have a simplified calculation, negative 16 over five times negative three over two squared. If we look at negative three over two squared, this is equivalent to negative three squared on the numerator and two squared on the denominator. This is because we can apply the exponent law ๐ฅ over ๐ฆ to the power of ๐ equals ๐ฅ to the power of ๐ over ๐ฆ to the power of ๐.

So letโs simplify what we can in our calculation. Negative three squared is nine and two squared is four. We can multiply fractions by multiplying the numerators and denominators but before that observe that we can simplify this. Four is a common factor of both negative 16 and four. So we calculate negative four times nine, which is negative 36. And five times one gives us five. So our final answer is negative 36 over five. It would also have been valid to give our answer as negative seven and one-fifth.

We can now summarize what weโve learned in this video. We saw that there are a number of laws we can use when working with exponents. The first two laws that we saw covered what happens when we multiply values with exponents and when we divide values with exponents. We saw the law covering what happens when we take the power of a power. In this case, we would multiply the exponents.

The final law that we saw covers what happens when we have a fraction raised to the power of ๐. This would be equivalent to the numerator and denominator both raised to the power of ๐. And one final point is to remember that if weโre working with mixed numbers raised to a power, then always make sure we change those to improper fractions.

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