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Simplify. Assume all variables are positive. $\newline$ $\frac{r^{\frac{2}{3}}}{r^{\frac{7}{3}}}$ $\newline$ 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{r^{\frac{2}{3}}}{r^{\frac{7}{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

______

Identify Expression: We have the expression: $\newline$ $r^{\frac{2}{3}} / r^{\frac{7}{3}}$ $\newline$ Which operation will be applied with the exponents? $\newline$ When dividing powers with the same base, the exponents are subtracted. $\newline$ So, we will use the subtraction operation with the exponents.
Apply Exponent Rule: Subtract the exponents of $r$ : $r^{\frac{2}{3}} / r^{\frac{7}{3}} = r^{(\frac{2}{3} - \frac{7}{3})}$ Perform the subtraction: $r^{(\frac{2}{3} - \frac{7}{3})} = r^{-\frac{5}{3}}$
Simplify Negative Exponent: Simplify the expression with a negative exponent: $r^{(-5/3)}$ can be rewritten as $1 / r^{(5/3)}$ This gives us a positive exponent, as required.

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