In Exercises 91–100, simplify using properties of exponents. (3y^...

Question

In Exercises 91–100, simplify using properties of exponents. (3y^1/4)^3/y^1/12

Accepted Answer

Welcome back, I'm so glad you're here. We've got an expression, it is in the numerator we have five X to the 1/5 and that whole numerator is being raised to the fourth and that is over X to the 3/10. Where to simplify using the laws of exponents. There are many different ways you can go about doing this. The order isn't so important. So you might have done something totally different and arrive at the same answer. But here's how I'm going to do it. I'm going to take this four that the five times X to the 1/5 is being distributed or raised to and I am going to distribute that exponent. That four to the five into the X to the 1/5. We recall from previous lessons that when you've got an exponent raised to an exponent we're going to multiply so that five there it's like five to the first. So we're going to take the one times four. And so now this is five to the fourth and that X is going to be X to the 1/5 times four up in the exponent. And we can just carry over this X to the 3/10. Now we can do the applicable multiplication in the numerator, five to the fourth times X to the 1/5 times four. That is 4/5 All over X to the 3/10. What can we do next? I am going to combine these like terms here these Xs have a common base and we recall from previous lessons that when we have a common base and the terms are being divided, then the exponents are going to be subtracted. So we've got five to the fourth times X. Raised to the 4/5 so minus 3/10. Now I'm going to rewrite that so that we've got a common denominator for our two fractions instead of 4/5 I'm going to rate 8/10 minus 3/10. And that is the same as five to the fourth times X to the 5/10 or five to the fourth times X to the one half. We're almost there. 5 to the fourth. Okay. What is that? That is 625 and X to the one half when we look at our answer choices and they don't have any X to the one half. And we recall that X to the one half is the same as the square root of X. And now that matches with our answer choice. C. While done, we'll catch you on the next one.

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