Simplify. Assume all variables are positive. $\newline$ $\frac{u^{\frac{1}{3}}}{u^{\frac{7}{3}} \cdot u^{\frac{4}{3}}}$ $\newline$ 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{u^{\frac{1}{3}}}{u^{\frac{7}{3}} \cdot u^{\frac{4}{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

______

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. $\newline$ $u^{\frac{7}{3}} \times u^{\frac{4}{3}} = u^{\frac{7}{3} + \frac{4}{3}}$
Add exponents in denominator: Add the exponents in the denominator. $\newline$ $\frac{7}{3} + \frac{4}{3} = \frac{11}{3}$ $\newline$ So, $u^{\frac{7}{3}} \times u^{\frac{4}{3}} = u^{\frac{11}{3}}$
Rewrite with combined exponent: Rewrite the original expression with the combined exponent in the denominator. $\frac{u^{\frac{1}{3}}}{u^{\frac{11}{3}}}$
Apply property of exponents: Apply the property of exponents that states when you divide like bases, you subtract the exponents. $\newline$ $u^{\frac{1}{3}} / u^{\frac{11}{3}} = u^{\frac{1}{3} - \frac{11}{3}}$
Subtract exponents: Subtract the exponents. $\newline$ $\frac{1}{3} - \frac{11}{3} = -\frac{10}{3}$ $\newline$ So, $u^{\frac{1}{3}} / u^{\frac{11}{3}} = u^{-\frac{10}{3}}$
Rewrite as positive exponent: Since we want the exponent to be positive, we can rewrite $u^{(-10/3)}$ as $1/u^{(10/3)}$ .