Simplify. Assume all variables are positive. $\newline$ $\frac{v^{\frac{2}{3}}}{v^{\frac{7}{3}} \cdot v^{\frac{5}{3}}}$ $\newline$ 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. $\newline$ ______

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Precalculus

Simplify expressions involving rational exponents

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Q. Simplify. Assume all variables are positive.

\newline

\frac{v^{\frac{2}{3}}}{v^{\frac{7}{3}} \cdot v^{\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

______

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. $\newline$ $v^{\frac{7}{3}} \times v^{\frac{5}{3}} = v^{\frac{7}{3} + \frac{5}{3}}$
Add Exponents: Add the exponents in the denominator. $\newline$ $\frac{7}{3} + \frac{5}{3} = \frac{12}{3}$ $\newline$ $v^{\frac{7}{3} + \frac{5}{3}} = v^{\frac{12}{3}}$
Simplify Exponent: Simplify the exponent in the denominator. $\newline$ $\frac{12}{3} = 4$ $\newline$ $v^{\frac{12}{3}} = v^4$
Rewrite Expression: Rewrite the original expression with the simplified denominator. $\frac{v^{\frac{2}{3}}}{v^{4}}$
Apply Exponent Property: Apply the property of exponents that states when you divide like bases, you subtract the exponents. $\newline$ $v^{\frac{2}{3}} / v^4 = v^{\frac{2}{3} - 4}$
Convert Whole Number: Convert the whole number $4$ to a fraction with the same denominator as $\frac{2}{3}$ to subtract the exponents. $\newline$ $4 = \frac{12}{3}$ $\newline$ $v^{\frac{2}{3} - 4} = v^{\frac{2}{3} - \frac{12}{3}}$
Subtract Exponents: Subtract the exponents. $\newline$ $\frac{2}{3} - \frac{12}{3} = -\frac{10}{3}$ $\newline$ $v^{\frac{2}{3} - \frac{12}{3}} = v^{-\frac{10}{3}}$
Use Negative Exponent Property: Since we want the exponent to be positive, we can use the property of exponents that states a negative exponent means the reciprocal of the base raised to the positive exponent. $\newline$ $v^{(-10/3)} = 1/v^{(10/3)}$