Simplify. Assume all variables are positive. $\newline$ $\frac{z^{\frac{1}{2}}}{z^{\frac{3}{2}} \cdot z^{\frac{3}{2}}}$ $\newline$ 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. $\newline$ ______

Full solution

Q. Simplify. Assume all variables are positive.

\newline

\frac{z^{\frac{1}{2}}}{z^{\frac{3}{2}} \cdot z^{\frac{3}{2}}}

\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

______

Given Expression: We are given the expression $z^{\frac{1}{2}}/(z^{\frac{3}{2}} \cdot z^{\frac{3}{2}})$ . To simplify this expression, we will use the properties of exponents to combine the exponents in the denominator and then simplify the fraction.
Combine Exponents in Denominator: First, let's combine the exponents in the denominator using the property of exponents that states $a^{m} \times a^{n} = a^{m+n}$ . We have $z^{\frac{3}{2}} \times z^{\frac{3}{2}}$ which equals $z^{\frac{3}{2} + \frac{3}{2}}$ .
Simplify Denominator: Adding the exponents in the denominator, we get $z^{\frac{3}{2} + \frac{3}{2}} = z^{\frac{6}{2}}$ . Simplifying $\frac{6}{2}$ gives us $z^3$ .
Combine Exponents in Numerator: Now we have the expression $z^{1/2} / z^3$ . To simplify this, we use the property of exponents that states $a^{m} / a^{n} = a^{m-n}$ when $m > n$ . Here, we have $z^{1/2 - 3}$ .
Simplify Numerator: Subtracting the exponents, we get $z^{1/2 - 3} = z^{1/2 - 6/2} = z^{-5/2}$ . Since we want the exponent to be positive, we can write this as $1/z^{5/2}$ .
Final Simplification: The expression is now simplified to $\frac{1}{z^{\frac{5}{2}}}$ , which is in the form $\frac{A}{B}$ as required, where $A$ is $1$ and $B$ is $z^{\frac{5}{2}}$ . There are no variables in common in the numerator and denominator, and the exponent is positive.