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Simplify. Assume all variables are positive. $\newline$ $\frac{d^{\frac{1}{4}}}{d^{\frac{9}{4}}}$ $\newline$ 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{d^{\frac{1}{4}}}{d^{\frac{9}{4}}}

\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: $\newline$ $d^{\frac{1}{4}}/d^{\frac{9}{4}}$ $\newline$ To simplify this expression, we need to apply the rule for dividing powers with the same base, which states that we should subtract the exponents.
Subtract Exponents of $d$ : Subtract the exponents of $d$ to combine the expression into a single power of $d$ :
$\frac{d^{1/4}}{d^{9/4}} = d^{(1/4)-(9/4)}$
Perform the subtraction:
$d^{(1/4)-(9/4)} = d^{-8/4}$
Simplify Exponent by Division: Simplify the exponent by dividing $-8$ by $4$ : $d^{(-8/4)} = d^{(-2)}$
Rewrite Exponent as $1/d^2$ : Since we want the exponent to be positive, we rewrite $d^{-2}$ as $1/d^2$ : $d^{-2} = 1/d^2$

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