Simplify. Assume all variables are positive. $\newline$ $\frac{w^{\frac{1}{3}}}{w^{\frac{8}{3}} \cdot w^{\frac{5}{3}}}$ $\newline$ 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{w^{\frac{1}{3}}}{w^{\frac{8}{3}} \cdot w^{\frac{5}{3}}}

\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

______

Write Given Expression: Write down the given expression. $\newline$ We have the expression $\frac{w^{\frac{1}{3}}}{w^{\frac{8}{3}} \cdot w^{\frac{5}{3}}}$ .
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. $\newline$ $w^{\frac{8}{3}} \times w^{\frac{5}{3}} = w^{\frac{8}{3} + \frac{5}{3}} = w^{\frac{13}{3}}$ .
Rewrite Expression: Rewrite the original expression with the combined exponent in the denominator. The expression now is $w^{\frac{1}{3}} / w^{\frac{13}{3}}$ .
Apply Exponent Property: Apply the property of exponents that states when you divide like bases, you subtract the exponents. $\newline$ $w^{\frac{1}{3}} / w^{\frac{13}{3}} = w^{\frac{1}{3} - \frac{13}{3}} = w^{-\frac{12}{3}}$ .
Simplify Exponent: Simplify the exponent. $w^{(-12/3)} = w^{(-4)}$ .
Rewrite with Positive Exponent: Since we want only positive exponents, we can rewrite $w^{-4}$ as $1/w^4$ .