Question

How do you simplify \[8{x^{ - 1}}{y^{ - 3}}\] and write it using only positive exponents?

Answer 1

Hint: First, we have to figure out what is asked and what we have to find out. According to the question, we have to modify the given expression which contains only positive exponents. For that we have to use certain properties of exponents.
Formula used: \[{a^{ - n}} = \dfrac{1}{{{a^n}}}\]

Complete step by step solution:
We can see that there is a negative exponent in the question. When we write a negative exponent, we assume that the exponent is a negative integer. It says that the base is divided by a certain number from the question or divided by \[1\] , exponent number of times.
According to the question, it is given that the \[ - 1\] exponent is applied to \[x\] , and the \[ - 3\] exponent is applied to \[y\] . It says that \[x\] is divided by \[8\] , \[1\] number of times, and \[y\] is divided with \[8\] , \[3\] number of times. Now, we have to see how we will change or modify the question by using the exponent properties. We have a formula related to negative exponent, and it is:
 \[{a^{ - n}} = \dfrac{1}{{{a^n}}}\]
When there is a negative exponent in a question, then it tells us that we have to find the reciprocal of the base.
So, we will apply this formula in our question as well. We will shift the negative exponent in the denominator. Here, in the question, the \[x\] and its exponent is going to be shifted to the denominator, and similarly \[y\] and its exponent is also going to be shifted to the denominator. Here, \[8\] remains in the numerator as it is a only number and does not have any negative exponent. So, we get:
 \[ = \dfrac{8}{{{x^1}{y^3}}}\]
We know that any base having the exponent \[1\] gives the base only as a result. So, here also we can see that \[x\] is having the exponent as \[1\] . So, it’s answer will be \[x\] only. So, we get:
 \[ = \dfrac{8}{{x{y^3}}}\]
Therefore, our final result is that \[8{x^{ - 1}}{y^{ - 3}} = \dfrac{8}{{x{y^3}}}\] .
So, the correct answer is “$\dfrac{8}{{x{y^3}}}$”.

Note: When it comes to a positive exponent, it is similar to a negative exponent only. It says that the base is multiplied with a certain number from the question or multiplied by \[1\] , exponent number of times. In case of positive exponents, we do not shift the base and the exponent in the denominator.