In Exercises 111–114, simplify each expression. Assume that all v...

Question

In Exercises 111–114, simplify each expression. Assume that all variables represent positive numbers. (x^−5/4y^1/3x^−3/4)^−6

Accepted Answer

Welcome back. I'm so glad you're here. We've got an expression here, I'll rewrite it. We've got a raised to the negative seven thirds times B raised to the four thirds All over B raised to the negative 5/3. And this whole thing Is being raised to the negative 3rd and all variables are greater than zero. That just means we're not dealing with zeros where we have to worry about the domain in our fraction. So there's a lot of different ways you could solve this and work with this. I am going to start by clearing this bracket and distributing this negative three exponent to each of our three terms. And when you've got an exponent raised to an exponent from our rules of exponents, we recall that means it will be multiplied the exponents will be multiplied. So doing this first one, A Raised to the -7/3 is now being multiplied by -3. That's times B Raised to the 4/3, which is now being multiplied by -3. All over B. To the negative five thirds. Also now being multiplied by negative three and we've cleared that bracket And distributed our -3 exponent. So we can do a little bit of canceling out here. The threes here cancel out this is just going to be a negative one. The threes here cancel out this will just be a negative one and the threes here cancel out and this will just be a negative one. Now we can multiply, we've got a race to the negative seven times negative one. That's eight of the seventh, times B raised to the four times negative one. That's negative four. All over. B raised to the -5 times negative one which is five. Now the only thing we need to do here Is note that this negative in our exponent here be raised to the -4, tells us that this term has to get flipped to the other side of the denominator to clear the negative. So we've got a race to the seventh on top and we've got be raised to the fifth times B raised to the fourth on the bottom. We know that when two terms with the same base are being multiplied and they have exponents. We can add those exponents together. So this will simplify to a race to the seventh over. Be raised to the five plus four is nine. We look at our answer choices and this matches with answer choice. A. Well done. We'll catch you on the next one.

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