Simplify each rational expression. Assume all variable expression...

Question

Simplify each rational expression. Assume all variable expressions represent positive
real numbers. (Hint: Use factoring and divide out any common factors as a first step.) [2(2x-3)^1/3 - (x-1)(2x-3)^-2/3] / [(2x-2)^-2/3]

Accepted Answer

Hello everyone. We're gonna simplify the expression four times three Y plus two raised the 1/5 minus Y plus five times three Y plus two raised to the negative 4/5 all over three Y plus two raised the 4/5 by factoring and eliminating any common factors that we find. First thing I'm noticing is that I have a negative exponent on my 4/5 which means that I need to rewrite that as a denominator of a fraction. I recall that I only move the piece that is officially on so it's not going to move the Y plus five just the three Y plus two. So my numerator now looks like four times three Y plus two to the 1/5 -Y. Plus five over three Y plus two to the 4/5. And then that's all over three Y plus two to the 4/5. Next I'm gonna make a common denominator for my numerator. So I can try to simplify this fraction on the top. So I'm going to multiply top and bottom of that first part by three Y plus two to the 4/5. So when I do that I get four times three Y plus two to the 1/5 Times three wide plus to the 4/5 over three Y plus two to the 4/5 minus Y. Plus 5/3 Y plus two To the 4/5. All of that over three Y plus two to the 4/5. Alright next I'm going to take the numerator and make it one fraction. So also when I do that I'm going to combine my three y plus two to the 1/5 and my three Y plus two to the 4/5 because I recall that if I have exponents and they have the same base when I multiply them I can add their exponents. So I think that'll help clean this up a little bit too. So four times. Well if I add the exponents of 1/5 and 4/5 I get 5/5 which is one which is the same thing as just saying I have three Y plus two and now it's minus Y plus five over three Y plus two. The 4/5 Over three Y plus two To the 4/ scroll down a little give us a little bit more space. I want to simplify the numerator of my numerator so I'm going to distribute the four into the first set of parentheses, giving myself 12 Y plus eight and I'm gonna distribute the minus sign into the second set of parentheses. So minus y minus five over three y plus two, Raise the 4/5 over three Y plus two, raised to the 4/5 Finishing cleaning up that numerator. I get 11 y plus three Over three Y Plus two. The 4/5. And I recognize that it's the same as saying it's divided by three Y plus two to the 4/5. This helps me because now that I have it set up as a fraction. I can treat this as if it's over one. I can use my fraction rules to simplify this and hopefully finish it up pretty soon. So when I have fractions that I'm dividing, that means I'm multiplying by the reciprocal. So my first fraction stays the same. 11 y plus three over three Y plus two to the 4/5. My second fraction I make it multiplication in the middle and I flip it over so I get one over three Y plus two to the 4/5. And I'm gonna multiply straight across. Well I guess that'll go right here for now. 11 Y plus three times one is 11 Y plus three in the denominator. They are the exact same. So it's the same as saying it's squared. So that would be like saying I have three y plus two. The 4/5 to the 2nd power. And when I raise the power to a power I multiply my exponents. So I have 11 Y plus three over three Y plus two two. Now the 8/5 and that looks like as simplified as it gets. So that is my solution. I'm going to rewrite it up here so I can check it with our possible answer choices. So we have as our answer 11 Y plus three over three Y plus two to the 8/5. And when I scroll back up that looks like it is answer choice. Let's see it's answer choice. C have a good day

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Simplify each rational expression. Assume all variable expressions represent positive real numbers. (Hint: Use factoring and divide out any common factors as a first step.) [2(2x-3)^1/3 - (x-1)(2x-3)^-2/3] / [(2x-2)^-2/3]