Simplify. Assume all variables are positive. $\newline$ $\frac{s^{\frac{3}{2}}}{s^{\frac{5}{2}} \cdot s^{\frac{1}{2}}}$ $\newline$ 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. $\newline$ ______ $\newline$

Full solution

Q. Simplify. Assume all variables are positive.

\newline

\frac{s^{\frac{3}{2}}}{s^{\frac{5}{2}} \cdot s^{\frac{1}{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

______

\newline

Identify Given Expression: Identify the given expression and the rule for dividing exponents with the same base. $\newline$ The expression is $s^{\frac{3}{2}} / (s^{\frac{5}{2}} \cdot s^{\frac{1}{2}})$ . When dividing exponents with the same base, we subtract the exponents.
Apply Division Rule: Apply the rule for dividing exponents to the given expression. $\newline$ $s^{\frac{3}{2}} / (s^{\frac{5}{2}} \cdot s^{\frac{1}{2}}) = s^{\frac{3}{2}} / s^{\frac{5}{2} + \frac{1}{2}}$
Add Exponents: Add the exponents in the denominator. $\newline$ $\frac{5}{2} + \frac{1}{2} = \frac{6}{2}$ $\newline$ $s^{\frac{3}{2}} / s^{\frac{5}{2} + \frac{1}{2}} = s^{\frac{3}{2}} / s^{\frac{6}{2}}$
Simplify Denominator: Simplify the exponent in the denominator. $\newline$ $\frac{6}{2} = 3$ $\newline$ $\frac{s^{\frac{3}{2}}}{s^{\frac{6}{2}}} = s^{\frac{3}{2} - 3}$
Subtract Exponents: Subtract the exponents in the numerator and the denominator. $\newline$ $\frac{3}{2} - 3 = \frac{3}{2} - \frac{6}{2} = -\frac{3}{2}$ $\newline$ $s^{\frac{3}{2}} / s^3 = s^{\frac{3}{2} - 3} = s^{-\frac{3}{2}}$
Rewrite with Positive Exponent: Since the problem asks for positive exponents, rewrite the expression with a positive exponent. $\newline$ $s^{(-\frac{3}{2})} = \frac{1}{s^{(\frac{3}{2})}}$