# Thread: Simplify Rational Expression with Fractional Exponents

1. ## Simplify Rational Expression with Fractional Exponents

I'm a bit stuck on this one.

Simplify:

$\displaystyle \frac{\frac{3}{x^{\frac{1}{2}}}+\frac{1}{x^{7}}} {6-\frac{2}{x}}$

2. Why does the exponent type change anything. Just proceed as usual.

Add fractions by finding a common denominator. $\displaystyle \frac{3}{x^{\frac{1}{2}}}+\frac{1}{x^7}\;=\;\frac{ 3\cdot x^{\frac{13}{2}}+1}{x^7}$

3. You can also begin by multiplying each term by $\displaystyle x^7$. This will turn the complex fraction into a simple fraction.

4. Ok. So the common denominator for the top is x^(13/2) because the base x remains the same and the exponents are added together. So that give x^14/2 which is the same as x^7.

5. I think I'm on the wrong path here:

$\displaystyle \frac{3x^{\frac{15}{2}}+x}{6x^{8}-2x^{7}}$

6. What you have is actually correct. I'm just not sure why you multiplied each term by $\displaystyle x^8$ instead of $\displaystyle x^7$.

7. I just encountered another problem that is very similar to the one I've been having trouble with. All I got to do is simplify it. It seems that whatever I try leads me to the wrong answer. I would appreciate if someone can show me step by step how to handle this type of problem. Thanks

This is an image of where I get stuck:

8. Dividing each term by x just makes the fraction more complicated. You want to multiply each term by $\displaystyle x^8$.

Note that $\displaystyle x^{-8}=\frac{1}{x^8}$.

9. I know that I need to divide by x because I'm using coursecompass and it shows to do that. The other steps are skipped. I'm trying to find the limit of that function as x approaches infinity. I'm stuck on the algebra of something that shouldn't take me more than a few minutes to figure out. That's why I was requesting a step by step show of the algebra because that's what coursecompass is skipping.

10. You didn't say you were taking a limit. Just take the highest power of x on top and on bottom, divide and take the limit:

$\displaystyle \frac{5\sqrt{x}}{7x}=\frac{5}{7\sqrt{x}}$. This approaches 0.

11. I know I didn't mention I was taking a limit because I thought all I had to do was simplify and then evaluate. I realized soon after my last post that after dividing the expression by the highest power of x (which is just x in this case), I then evaluate. Thanks anyway for the help.

12. Dividing by the highest power of x is not simplifying the complex fraction. But it is the best way to evaluate the limit.