Simplify. Assume all variables are positive. $\newline$ $\frac{x^{1/2}}{x^{5/2} \cdot x^{3/2}}$ $\newline$ 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{x^{1/2}}{x^{5/2} \cdot x^{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: Combine the exponents in the denominator using the property of exponents that states when you multiply like bases, you add the exponents. $\newline$ $x^{\frac{5}{2}} \times x^{\frac{3}{2}} = x^{\frac{5}{2} + \frac{3}{2}}$
Add exponents in denominator: Add the exponents in the denominator. $\newline$ $\frac{5}{2} + \frac{3}{2} = \frac{8}{2}$ $\newline$ $\frac{8}{2} = 4$ $\newline$ So, $x^{\frac{5}{2}} \times x^{\frac{3}{2}} = x^4$
Rewrite with simplified denominator: Rewrite the original expression with the simplified denominator. $\frac{x^{\frac{1}{2}}}{x^4}$
Apply division property of exponents: Apply the property of exponents that states when you divide like bases, you subtract the exponents. $\newline$ $x^{\frac{1}{2}} / x^4 = x^{\frac{1}{2} - 4}$
Convert whole number to fraction: Convert the whole number $4$ into a fraction with the same denominator as $\frac{1}{2}$ to subtract the exponents. $\newline$ $4 = \frac{8}{2}$ $\newline$ So, $x^{\frac{1}{2} - 4} = x^{\frac{1}{2} - \frac{8}{2}}$
Subtract exponents: Subtract the exponents. $\newline$ $\frac{1}{2} - \frac{8}{2} = -\frac{7}{2}$ $\newline$ So, $x^{\frac{1}{2} - 4} = x^{-\frac{7}{2}}$
Use property to make exponent positive: Since we want the exponent to be positive, we can use the property of exponents that states $x^{-a} = \frac{1}{x^{a}}$ . $\newline$ $x^{-\frac{7}{2}} = \frac{1}{x^{\frac{7}{2}}}$