Simplify. Assume all variables are positive. $\newline$ $\frac{z^{\frac{3}{4}}}{z^{\frac{9}{4}} \cdot z^{\frac{3}{4}}}$ $\newline$ 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{z^{\frac{3}{4}}}{z^{\frac{9}{4}} \cdot z^{\frac{3}{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 Given Expression: Identify the given expression and the operation to be performed. $\newline$ The expression is $z^{\frac{3}{4}}$ divided by $(z^{\frac{9}{4}} \cdot z^{\frac{3}{4}})$ . $\newline$ We need to simplify this expression using the properties of exponents.
Apply Exponent Property for Multiplication: Apply the property of exponents for multiplication within the denominator. According to the property $a^m \times a^n = a^{(m+n)}$ , we can add the exponents when multiplying like bases. So, $z^{\frac{9}{4}} \times z^{\frac{3}{4}} = z^{\frac{9}{4} + \frac{3}{4}}$ .
Calculate Exponent Sum: Calculate the sum of the exponents in the denominator. $\newline$ $\frac{9}{4} + \frac{3}{4} = \frac{12}{4}$ . $\newline$ So, $z^{\frac{9}{4}} \times z^{\frac{3}{4}} = z^{\frac{12}{4}}$ .
Simplify Denominator: Simplify the exponent in the denominator. $\newline$ $\frac{12}{4}$ simplifies to $3$ . $\newline$ So, $z^{\frac{9}{4}} \cdot z^{\frac{3}{4}} = z^3$ .
Rewrite Expression: Rewrite the original expression with the simplified denominator. The expression now is $\frac{z^{\frac{3}{4}}}{z^3}$ .
Apply Exponent Property for Division: Apply the property of exponents for division. According to the property $a^m / a^n = a^{(m-n)}$ , we can subtract the exponents when dividing like bases. So, $z^{3/4} / z^3 = z^{(3/4 - 3)}$ .
Calculate Exponent Difference: Calculate the difference of the exponents. $\newline$ $\frac{3}{4} - 3$ is the same as $\frac{3}{4} - \frac{12}{4}$ , which equals $-\frac{9}{4}$ . $\newline$ So, $z^{\frac{3}{4}} / z^3 = z^{-\frac{9}{4}}$ .
Rewrite with Positive Exponent: Since we want all exponents to be positive, rewrite the expression with a positive exponent. $\newline$ According to the property $a^{-n} = \frac{1}{a^n}$ , we can write $z^{-\frac{9}{4}}$ as $\frac{1}{z^{\frac{9}{4}}}$ . $\newline$ So, $\frac{z^{\frac{3}{4}}}{z^3} = \frac{1}{z^{\frac{9}{4}}}$ .
Check Final Expression: Check if the final expression answers the question prompt. $\newline$ The final expression is 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 the answer are positive.