Question Video: Simplifying an Expression Involving Exponentials Mathematics

Simplify ((36)^(1/4) × (21)² × (8)^(1/5))/((486)^(1/10) × (42)³).

06:55

Video Transcript

Simplify 36 raised to the power of one-quarter times 21 squared multiplied by eight to the power of one-fifth all divided by 486 to the power of one over 10 times 42 cubed.

In this question, we’re asked to simplify an expression involving the products and quotient of exponential expressions. And to do this, we’re going to need to use the laws of exponents. We could do this all at once. However, it’s easier to do this factor by factor. This makes it much less likely that we’ll make a mistake. Let’s start with the first factor in the numerator. That’s 36 to the power of one-quarter.

To simplify this, we’re going to first factor the base 36 into primes. The reason for this is we need to simplify an entire expression involved in the products and quotients of exponential expressions. And to use the laws of exponents to simplify an expression, we either need the exponents to be the same or the bases to be the same. So we should try and make the bases of each of these factors as simple as possible. This is why we find the prime factorization of the bases.

We see that 36 is six squared and six is two times three. So we can write 36 as two squared multiplied by three squared, which means we can rewrite this factor as two squared times three squared all raised to the power of one-quarter. We want to write this in terms of exponential expressions with bases two and three. So to do this, we’re going to need to use the laws of exponents. Since we’re raising a product to an exponent, we’ll use the result 𝑎 times 𝑏 all raised to the 𝑛th power is equal to 𝑎 to the 𝑛th power multiplied by 𝑏 to the 𝑛th power. Our value of 𝑎 is two squared. Our value of 𝑏 is three squared. And our value of 𝑛 is one-quarter. This gives us two squared to the power of one-quarter multiplied by three squared to the power of one-quarter.

Finally, to write these in an exponential form, we need to note that we’re raising an exponential expression to another exponent. And we know when we do this, we take the product of the exponents. 𝑎 to the power of 𝑛 all raised to the power of 𝑚 is equal to 𝑎 to the power of 𝑛 times 𝑚. And since two times one-quarter is one-half, we can simplify this to get two to the power of one-half times three to the power of one-half.

And it’s worth noting here we could rewrite this by using radicals. However, we’re going to use this to simplify the expression given to us in the question. And this means we’re probably going to need to apply laws of exponents, which means we’re going to need to use arithmetic on the exponents. So it’s easier to leave these as fractions. Let’s now move on to the second factor in the numerator. That’s 21 squared. We can follow a very similar process to rewrite this factor. First, we find the prime factorization of 21. It’s equal to three times seven.

Now, we once again apply our laws of exponents to distribute the exponent over the product. Three times seven all squared is equal to three squared times seven squared. Now we can apply a similar process to the third factor in our numerator. First, we factor the base of eight into primes, which we find is two cubed, meaning we can rewrite this as two cubed all raised to the power of one-fifth. Then, we can use one of our laws of exponents to simplify this to an exponential expression with base two. Two to the power of three all raised to the power of one-fifth is two to the power of three times one-fifth, or two to the power of three over five.

Let’s now move on to the first factor in our denominator. And we’ll follow the same process as we did for the other factors. First, we need to factor 486 into primes. If we do this, we’ll see that it’s equal to two times three to the fifth power. Therefore, this factor simplifies to give us two times three to the fifth power all raised to the power of one over 10. Now, since we’re raising a product to an exponent, we can use our laws of exponents to distribute the exponent over the product. We get two to the power of one over 10 multiplied by three to the fifth power raised to the power of one over 10.

And now we can simplify this further by noticing we’re raising three to the fifth power and raising this all to the power of one over 10. By our laws of exponents, this will be equal to three to the power of five times one over 10. And five over 10 is one-half. So this factor simplifies to give us two to the power of one over 10 multiplied by three to the power of one-half.

Finally, let’s move on to the last factor in our denominator. First, we factor the base 42 into primes to get two times three times seven. Then, we use our laws of exponents to distribute the exponent over our product. We get two cubed times three cubed times seven cubed. We’re now going to need to substitute all of these into the expression given to us in the question. To do this, let’s start by clearing some space and keeping track of the simplifications we’ve already made. Now, all we need to do is substitute these into the expression given to us in the question. This then gives us the following expression. We have a product and quotient of exponential expressions where the bases are all prime numbers and the exponents are rational numbers.

We can simplify this in a lot of different ways by using the laws of exponents. And it’s personal preference which way we might want to do this. We’re going to start by rewriting this expression so that we group all of the factors with like bases. For example, in the numerator we have two factors with base two. That’s two to the power of one-half and two to the power of three over five. And in our denominator, we have two factors with base two: two to the power of one over 10 and two cubed. So we can take all of these factors out to give us the following expression. We then need to multiply this by the rest of the expression, and we can take out all of the factors with base three. Then finally, we multiply this by the remaining factors with base seven. This gives us the following expression.

And it’s not necessary to do this step. However, it does make it a lot easier to see how we need to simplify this expression. First, we see we’re taking the product of exponential expressions with the same base. And to do this, we need to add the exponents together. 𝑎 to the power of 𝑛 times 𝑎 to the power of 𝑚 is equal to 𝑎 to the power of 𝑛 plus 𝑚. We could use this to simplify the numerators and denominators. However, we could also use the fact when we take the quotient of exponential expressions with the same base, we find the difference in the exponents. 𝑎 to the power of 𝑛 divided by 𝑎 to the power of 𝑚 is 𝑎 to the power of 𝑛 minus 𝑚. This means we can apply both of these steps at once. We need to add the exponents in the numerator and then subtract the exponents in the denominator.

But remember, it is personal preference. We can just simplify the numerators and denominators separately if we prefer. We’ll apply both steps at the same time. This means we need to add the exponents in the numerator together and subtract the exponents from the denominator. We get two to the power of one-half plus three-fifths minus one-tenth minus three. We can then apply the same method to simplify our second quotient. We get three to the power of one-half plus two minus one-half minus three. And for our third and final quotient, we have seven squared divided by seven cubed. We need to find the difference in the exponents. That’s seven to the power of two minus three.

Now, we just evaluate the expressions in the exponents. We get two to the power of negative two times three to the power of negative one multiplied by seven to the power of negative one. And we can simplify this further by using the following laws of exponents. 𝑎 to the power of negative 𝑛 is equal to one divided by 𝑎 to the 𝑛th power. This gives us one over two squared times one over three times one over seven, which we can evaluate. It’s equal to one divided by 84, which is our final answer. Therefore, we were able to simplify the exponential expression given to us in the question by using the laws of exponents. We showed it was equal to one divided by 84.

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