Simplify. Assume all variables are positive. $\newline$ $\frac{c^{\frac{3}{2}}}{c^{\frac{5}{2}} \cdot c^{\frac{1}{2}}}$ $\newline$ Write your answer in the form $A$ or $A/B$ , 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{c^{\frac{3}{2}}}{c^{\frac{5}{2}} \cdot c^{\frac{1}{2}}}

\newline

Write your answer in the form

A

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 laws of exponents to be used. $\newline$ The expression is $c^{\frac{3}{2}}/(c^{\frac{5}{2}} \cdot c^{\frac{1}{2}})$ . We will use the laws of exponents which state that when dividing like bases, we subtract the exponents and when multiplying like bases, we add the exponents.
Combine Exponents in Denominator: Combine the exponents in the denominator using the law of exponents for multiplication. $c^{\frac{5}{2}} \cdot c^{\frac{1}{2}} = c^{\frac{5}{2} + \frac{1}{2}} = c^{\frac{6}{2}} = c^3$ .
Divide Exponents: Divide the exponents using the law of exponents for division. $c^{\frac{3}{2}} / c^3 = c^{\frac{3}{2} - \frac{3}{1}} = c^{\frac{3}{2} - \frac{6}{2}} = c^{-\frac{3}{2}}$ .
Rewrite Negative Exponent: Since we want the exponent to be positive, we can rewrite $c^{(-3/2)}$ as $1/c^{(3/2)}$ .
Check Final Answer: Check to ensure the final answer has no variables in common in the numerator and denominator and that all exponents are positive. $\newline$ The final answer is $\frac{1}{c^{\frac{3}{2}}}$ , which meets the requirements.