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Simplify. Assume all variables are positive. $\newline$ $\frac{r^{\frac{4}{3}}}{r^{\frac{7}{3}}}$ $\newline$ 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{4}{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 operation: We have the expression: $\newline$ $r^{\frac{4}{3}} / r^{\frac{7}{3}}$ $\newline$ 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.
Subtract exponents: Subtract the exponents of $r$ : $\newline$ $r^{\frac{4}{3}} / r^{\frac{7}{3}} = r^{\left(\frac{4}{3} - \frac{7}{3}\right)}$ $\newline$ Perform the subtraction: $\newline$ $r^{\left(\frac{4}{3} - \frac{7}{3}\right)} = r^{-\frac{3}{3}}$
Perform subtraction: Simplify the exponent: $\newline$ $r^{(-3/3)} = r^{(-1)}$ $\newline$ Since we want the exponent to be positive, we rewrite $r^{(-1)}$ as: $\newline$ $r^{(-1)} = \frac{1}{r^{(1)}}$
Simplify exponent: The final simplified expression is: $\frac{1}{r}$ This is already in the form $A$ or $\frac{A}{B}$ , where $A$ and $B$ have no variables in common, and all exponents are positive.

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