# Thread: Simplify an expression involving fractional exponents by factoring

1. ## Simplify an expression involving fractional exponents by factoring

This one's driving me up the wall. It's not for homework, but I'm re-entering the math world after a liberal arts stint and am trying to be a perfectionist about it.

The expression: $\displaystyle (x^3 + 1)^2 (x - 1)^(-1/2) + 2x (x - 1)^(1/2) (x^2+1)$

(the -1/2 and 1/2 following the raised left parentheses are both supposed to be fractional exponents)

The book's answer: $\displaystyle [(x^2 + 1) (3x^2 - 2x + 1)] / sqrt(x-1)$

I'd just scan and post an image of the pages I've devoted solely to this problem if they were at all legible. Instead, I'll summarize my approaches to this problem:

I've tried putting (x-1)^1/2 in the denominator, then factoring everything out. I've tried putting that in the denominator, then adding like bases and factoring it out. I've tried just adding like bases to begin with, which totally prevented me from putting (x-1)^1/2 in the denominator. I've also tried squaring the whole problem to get rid of any (1/2) exponent and factoring it out. I even bought the Bagatrix "Algebra Solved!" software and had it try to simplify the problem, yet its answer doesn't match my books; its process matches my factoring process. I've double checked whether I've copied the initial problem correctly and I have.

P.S. Sorry about the formatting. Just blame my n00bhood.

2. Originally Posted by wolfe
This one's driving me up the wall. It's not for homework, but I'm re-entering the math world after a liberal arts stint and am trying to be a perfectionist about it.

The expression: $\displaystyle (x^3 + 1)^2 (x - 1)^(-1/2) + 2x (x - 1)^(1/2) (x^2+1)$

(the -1/2 and 1/2 following the raised left parentheses are both supposed to be fractional exponents)

The book's answer: $\displaystyle [(x^2 + 1) (3x^2 - 2x + 1)] / sqrt(x-1)$

I'd just scan and post an image of the pages I've devoted solely to this problem if they were at all legible. Instead, I'll summarize my approaches to this problem:

I've tried putting (x-1)^1/2 in the denominator, then factoring everything out. I've tried putting that in the denominator, then adding like bases and factoring it out. I've tried just adding like bases to begin with, which totally prevented me from putting (x-1)^1/2 in the denominator. I've also tried squaring the whole problem to get rid of any (1/2) exponent and factoring it out. I even bought the Bagatrix "Algebra Solved!" software and had it try to simplify the problem, yet its answer doesn't match my books; its process matches my factoring process. I've double checked whether I've copied the initial problem correctly and I have.

P.S. Sorry about the formatting. Just blame my n00bhood.
the original expression is ...

$\displaystyle (x^2 + 1)^2 (x - 1)^{-1/2} + 2x (x - 1)^{1/2} (x^2+1)$

... not an $\displaystyle x^3$ in the first factor.

common factors are $\displaystyle (x^2+1)$ and $\displaystyle (x-1)^{-1/2}$

$\displaystyle (x^2+1)(x-1)^{-1/2}[(x^2+1) + 2x(x-1)]$

$\displaystyle (x^2+1)(x-1)^{-1/2}[x^2+1 + 2x^2-2x]$

$\displaystyle (x^2+1)(x-1)^{-1/2}[3x^2-2x+1]$

$\displaystyle \displaystyle \frac{(x^2+1)(3x^2-2x+1)}{\sqrt{x-1}}$

3. Originally Posted by skeeter
the original expression is ...

$\displaystyle (x^2 + 1)^2 (x - 1)^{-1/2} + 2x (x - 1)^{1/2} (x^2+1)$

... not an $\displaystyle x^3$ in the first factor.

common factors are $\displaystyle (x^2+1)$ and $\displaystyle (x-1)^{-1/2}$

$\displaystyle (x^2+1)(x-1)^{-1/2}[(x^2+1) + 2x(x-1)]$

$\displaystyle (x^2+1)(x-1)^{-1/2}[x^2+1 + 2x^2-2x]$

$\displaystyle (x^2+1)(x-1)^{-1/2}[3x^2-2x+1]$

$\displaystyle \displaystyle \frac{(x^2+1)(3x^2-2x+1)}{\sqrt{x-1}}$
Oh sorry for the typo. You obviously know your stuff to have picked up on it!

The math makes beautiful sense to me once you rearranged it that first time. However, I'm having trouble understanding what's happening with all the motion. Particularly, where do the exponents 2 and 1/2 go in $\displaystyle (x^2 + 1)^2$ and $\displaystyle 2x(x-1)^{-1/2}$?

EDIT: I think I can reason that the 1/2 'vanishes' since, working backwards with the multiplication of exponents rule.. $\displaystyle (x-1)^{-1/2} * (x-1) = (x-1)^{-1/2} * (x-1)^{2/2} = (x-1)^{(-1/2) + (2/2)} = (x-1)^{1/2}$ .. however, it seems like in the original expression there are essentially 3 powers of $\displaystyle (x^2 + 1)$ (what with it being squared in the left side and multiplied later on the right).. and working backwards from your first rearrangement, I can only account for two powers (just the squared portion on the left side of the expression). My apologies if I'm being too deeply confusing.

4. Originally Posted by wolfe
Oh sorry for the typo. You obviously know your stuff to have picked up on it!

The math makes beautiful sense to me once you rearranged it that first time. However, I'm having trouble understanding what's happening with all the motion. Particularly, where do the exponents 2 and 1/2 go in $\displaystyle (x^2 + 1)^2$ and $\displaystyle 2x(x-1)^{1/2}$ ... fixed the last exponent
note that $\displaystyle (x^2+1)^2 = (x^2+1)(x^2+1)$

and $\displaystyle (x-1)^{1/2} = (x-1)^{-1/2} (x-1)$

5. OH. Hahaha. It just clicked. I rewrote that second portion of your advice as a radical and saw the light. Thanks for your patience skeeter. I'm officially in love with this site.