Simplify. Assume all variables are positive. $\newline$ $\frac{c^{5/2}}{c^{5/2} \cdot c^{3/2}}$ $\newline$ 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^{5/2}}{c^{5/2} \cdot c^{3/2}}

\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

______

Combine Exponents in Denominator: We start by looking at the expression $c^{\frac{5}{2}}/(c^{\frac{5}{2}} * c^{\frac{3}{2}})$ . To simplify, we can use the properties of exponents to combine the exponents in the denominator.
Simplify Denominator: Using the property of exponents that states $a^{m} \times a^{n} = a^{m+n}$ , we combine the exponents in the denominator: $\newline$ $c^{\frac{5}{2}} \times c^{\frac{3}{2}} = c^{\left(\frac{5}{2} + \frac{3}{2}\right)} = c^{\frac{8}{2}} = c^{4}.$
Rewrite Expression: Now we rewrite the original expression with the simplified denominator: $c^{\frac{5}{2}} / (c^{4})$ .
Use Exponent Property: Next, we use the property of exponents that states $a^{m} / a^{n} = a^{(m-n)}$ to simplify the expression further: $\newline$ $c^{($5$/$2$)} / c^{$4$} = c^{(($5$/$2$) - $4$)} = c^{(($5$/$2$) - ($8$/$2$))} = c^{($-3$/$2$)}.
Express with Positive Exponents: Since we are asked to write the answer with positive exponents and without variables in common in the numerator and denominator, we can express $c^{(-3/2)}$ as $1/c^{(3/2)}$.
Final Simplified Expression: The final simplified expression is $\frac{1}{c^{\frac{3}{2}}}$. This is the simplest form of the expression with positive exponents and no variables in common in the numerator and denominator.