Powers that are fractions?

• Jun 25th 2012, 10:17 PM
Powers that are fractions?
I have a question I'm not sure how to answer because I've never seen a power that is a fraction before. I need to simplify;

x^(1/3) times x^(2/3)

After some googling I think the fraction represents a cubed root, so the answer should be;

cubed root of (x times x^2)

Is that right?
• Jun 25th 2012, 10:42 PM
richard1234
Re: Powers that are fractions?
Yes.

$\displaystyle \sqrt[3]{x} \sqrt[3]{x^2} = \sqrt[3]{x*x^2} = x$ (for real x).
• Jun 25th 2012, 10:45 PM
Prove It
Re: Powers that are fractions?
Quote:

I have a question I'm not sure how to answer because I've never seen a power that is a fraction before. I need to simplify;

x^(1/3) times x^(2/3)

After some googling I think the fraction represents a cubed root, so the answer should be;

cubed root of (x times x^2)

Is that right?

The index laws hold for powers that are ANY numbers, so you can use \displaystyle \displaystyle \begin{align*} a^m \cdot a^n = a^{m + n} \end{align*} to simplify.

\displaystyle \displaystyle \begin{align*} x^{\frac{1}{3}} \cdot x^{\frac{2}{3}} &= x^{\frac{1}{3} + \frac{2}{3}} \\ &= x^1 \\ &= x \end{align*}
• Jun 25th 2012, 10:53 PM