In Exercises 55–78, use properties of rational exponents to simpl...

Question

In Exercises 55–78, use properties of rational exponents to simplify each expression. Assume that all variables represent positive numbers.16^¾ 16^¼

Accepted Answer

Hey, everyone in this problem, we're asked to simplify the following expression by using the properties of rational exponents. The expression we're given is 81 to the exponent 7/12 divided by 81 to the exponent 1/12. We're given four answer choices. Option A one half, option B 81 to the exponent two thirds. Option C nine and option D 81 to the exponent seven divided by 144. We're gonna start by rewriting what we were given. So we have 81 to the exponent 7/12 divided by 81 to the exponent 1/12. Now we're dividing two things that have the same base. So we're called the property of rational exponents that deals with the vision. If we have X to the exponent M divided by X to the exponent N, we can simplify this and write it as X to the exponent M minus N. So for dividing two things with the same base can turn this into a single term where the new exponent is the exponent of the original numerator minus the exponent of the original denominator. We apply this to our problem. We can write that we have 81 to the exponent 7/12 minus 1/12. Then we keep our original base 81. We take the exponent of the numerator 7/12 and subtract the exponent of the denominator 1/12. Now, we've got this down to a single term, but we want to simplify that exponent and we already have a common denominator of 12, 7/12 minus 1 12, gives us 6/12. We can simplify that fraction. We can divide both the numerator and denominator by six and we get 81 to the exponent one half. And now recall, we can write exponents as roots. So we can write 81 to the exponent one half as a root. How do we do that? Well, we're called that if we have X to the exponent one divided by N, this is equivalent to the NTH root of X. And so 81 to the exponent one half, our N value is two. And so this is equivalent to the square root of 81. And we know that the square root of 81 is equal to nine. And so the original expression we were given simplifies all the way down to the value nine. OK. So the correct answer is nine which corresponds with answer choice C, that's it for this one. Thanks everyone for watching. I hope this video helped

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In Exercises 55–78, use properties of rational exponents to simplify each expression. Assume that all variables represent positive numbers.16^¾ 16^¼

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