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Simplify. Assume all variables are positive. $\newline$ $\frac{u^{\frac{1}{3}}}{u^{\frac{2}{3}}}$ $\newline$ Write your answer in the form $A$ or $\frac{A}{B}$ , where $A$ and $B$ are constants or variable expressions that have no variables in common. All exponents in your answer should be positive. $\newline$ ______

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Division with rational exponents

Full solution

Q. Simplify. Assume all variables are positive.

\newline

\frac{u^{\frac{1}{3}}}{u^{\frac{2}{3}}}

\newline

Write your answer in the form

A

or

\frac{A}{B}

, where

A

and

B

are constants or variable expressions that have no variables in common. All exponents in your answer should be positive.

\newline

______

Apply Rule for Dividing Powers: We have the expression $(u^{\frac{1}{3}})/(u^{\frac{2}{3}})$ . We need to simplify this expression. When dividing powers with the same base, we subtract the exponents.
Perform Exponent Subtraction: Apply the rule for dividing powers with the same base to the expression. $\newline$ $\frac{u^{\frac{1}{3}}}{u^{\frac{2}{3}}} = u^{\left(\frac{1}{3}-\frac{2}{3}\right)}$
Rewrite with Positive Exponent: Perform the subtraction of the exponents. $\newline$ $u^{(\frac{1}{3})-(\frac{2}{3})} = u^{(-\frac{1}{3})}$
Rewrite with Positive Exponent: Perform the subtraction of the exponents. $\newline$ $u^{(\frac{1}{3})-(\frac{2}{3})} = u^{(-\frac{1}{3})}$ Since we want the exponent to be positive, we rewrite the expression with a positive exponent. $\newline$ $u^{(-\frac{1}{3})} = \frac{1}{u^{(\frac{1}{3})}}$

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