Simplify. Assume all variables are positive. $\newline$ $\frac{d^{\frac{5}{4}}}{d^{\frac{9}{4}} \cdot d^{\frac{5}{4}}}$ $\newline$ 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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Precalculus

Simplify expressions involving rational exponents

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Q. Simplify. Assume all variables are positive.

\newline

\frac{d^{\frac{5}{4}}}{d^{\frac{9}{4}} \cdot d^{\frac{5}{4}}}

\newline

Write your answer in the form

A

\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 and Apply Quotient of Powers: Identify the expression and apply the quotient of powers property. $\newline$ The expression is $d^{\frac{5}{4}} / (d^{\frac{9}{4}} \cdot d^{\frac{5}{4}})$ . According to the quotient of powers property, when dividing like bases, we subtract the exponents: $a^m / a^n = a^{m-n}$ .
Combine Terms Using Product of Powers: Combine the terms in the denominator using the product of powers property. $\newline$ The product of powers property states that when multiplying like bases, we add the exponents: $a^m \times a^n = a^{m+n}$ . $\newline$ So, $d^{\frac{9}{4}} \times d^{\frac{5}{4}} = d^{\frac{9}{4} + \frac{5}{4}} = d^{\frac{14}{4}}$ .
Apply Quotient of Powers to Numerator and Denominator: Apply the quotient of powers property to the numerator and the combined denominator. $\newline$ Now we have $d^{5/4} / d^{14/4}$ . Using the quotient of powers property, we subtract the exponents: $5/4 - 14/4 = -9/4$ . $\newline$ So, $d^{5/4} / d^{14/4} = d^{-9/4}$ .
Rewrite with Positive Exponent: Since we want the exponent to be positive, we can rewrite the expression with a positive exponent by taking the reciprocal of the base. $\newline$ $d^{(-\frac{9}{4})}$ is equivalent to $\frac{1}{d^{\frac{9}{4}}}$ .
Write Final Answer: Write the final answer in the form $A$ or $\frac{A}{B}$ , where $A$ and $B$ are constants or variable expressions that have no variables in common, and all exponents are positive. $\newline$ The final answer is $\frac{1}{d^{\frac{9}{4}}}$ , which is already in the correct form.