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Simplify. Assume all variables are positive. $\newline$ $\frac{z^{\frac{8}{5}}}{z^{\frac{13}{5}}}$ $\newline$ 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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Simplify expressions involving rational exponents

Full solution

Q. Simplify. Assume all variables are positive.

\newline

\frac{z^{\frac{8}{5}}}{z^{\frac{13}{5}}}

\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: $\newline$ $z^{\frac{8}{5}} / z^{\frac{13}{5}}$ $\newline$ Which operation will be applied with the exponents? $\newline$ When dividing powers with the same base, the exponents are subtracted.
Subtract Exponents: Now, let's simplify the expression by subtracting the exponents: $\newline$ $z^{\frac{8}{5}} / z^{\frac{13}{5}}$ $\newline$ = $z^{\frac{8}{5} - \frac{13}{5}}$ $\newline$ = $z^{-\frac{5}{5}}$
Use Negative Exponents: Since we want the exponent to be positive, we can rewrite the expression using the property of negative exponents: $\newline$ $z^{(-5/5)} = \frac{1}{z^{(5/5)}}$
Simplify Exponent: Now, simplify the exponent: $\frac{1}{z^{\frac{5}{5}}} = \frac{1}{z^1}$
Final Simplified Expression: Since $z^1$ is just $z$ , the final simplified expression is: $\frac{1}{z}$

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