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

______

Apply Quotient Rule: Write down the expression and apply the quotient rule for exponents. $\newline$ The quotient rule states that when dividing like bases with exponents, you subtract the exponents: $a^m / a^n = a^{(m-n)}$ . $\newline$ $u^{4/3} / (u^{7/3} \cdot u^{4/3}) = u^{4/3} / u^{(7/3 + 4/3)}$
Add Exponents: Add the exponents in the denominator. $\newline$ $u^{\frac{7}{3} + \frac{4}{3}} = u^{\frac{11}{3}}$ $\newline$ Now the expression is $\frac{u^{\frac{4}{3}}}{u^{\frac{11}{3}}}$ .
Subtract Exponents: Subtract the exponents in the denominator from the exponent in the numerator. $\newline$ $u^{\frac{4}{3}} / u^{\frac{11}{3}} = u^{\frac{4}{3} - \frac{11}{3}} = u^{-\frac{7}{3}}$
Take Reciprocal: Since we want all exponents to be positive, we can rewrite the expression with a positive exponent by taking the reciprocal of the base. $\newline$ $u^{-\frac{7}{3}} = \frac{1}{u^{\frac{7}{3}}}$
Write Final Answer: Write the final answer in the form $A$ or $A/B$ as requested. $\newline$ The final answer is $\frac{1}{u^{\frac{7}{3}}}$ , which is already in the form $A/B$ where $A$ and $B$ have no variables in common and all exponents are positive.