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.) [4(x^2- 1)^3 + 8x(x^2-1)^4] / [16(x^2-1)^3]

Accepted Answer

Hello everyone. We're going to simplify the rational expression. Six times a squared minus five raised the seventh power plus 24 A times a squared minus five raised to the eighth Power over 42. A squared minus five raised to the seventh power First. I recall that I can't break things up unless I factor them first. Looking at my numerator I want to see if there's anything common. I can factor out of both terms. Looking at the real number part 1st. I recognize that six goes evenly into both six and 24 so that's gonna be part of my greatest common factor that I'm factoring out. The other thing that I recognize both terms have is a squared minus 51 of them is to the seventh once to the eighth. That means I can factor seven out as an exponent and I'm going to have to leave one in that second term. So what that looks like is a squared minus five to the seventh power. And now what's left of my two terms for that first term? I took the whole term out. So I have a one For my second term, 24 divided by six is 4. I couldn't do anything with the a. So that stays put And then times a squared -5 because they took away seven from that exponent 8 -7 leaves me with one. Then in my denominator I'm going to write it just how it is and work from there. So that remains 42 times a squared -5. All raised to the 7th power. So now I can work on factoring things because I factored out the a squared minus five to the seventh power. I can cancel it out with the denominator so they can cancel right here. Next looking at the real number part, I have a six and a 42 and I'm trying to think if there's something that reduces both of them and I want to say it is 66 goes evenly into both of those numbers, six goes into six, once goes into 42 7 times. So now I'm gonna rewrite what I have left to double check, see if anything can be simplified. I have one plus four A times a squared -5 over seven and it doesn't look like there's anything else to factor out or cancel. But we do need to take this one step further and simplify the numerator. So we don't have parentheses so my one plus is going to stay put when I distribute my four A into my parentheses. So four A times eight of the second power is going to be 48 of the third power. 4, 8 times -5 gives me -20 a over seven. And that looks like as simplified as this is going to get and that appears to be answer choice B. So it is one plus 48 of the third minus 28/7. Just answer choice B. Have a nice day

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.) [4(x^2- 1)^3 + 8x(x^2-1)^4] / [16(x^2-1)^3]

Key Concepts

Rational Expressions

Factoring

Common Factors

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