Simplify. Assume all variables are positive. $\newline$ $\frac{x^{\frac{2}{3}}}{x^{\frac{5}{3}} \cdot x^{\frac{1}{3}}}$ $\newline$ 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{x^{\frac{2}{3}}}{x^{\frac{5}{3}} \cdot x^{\frac{1}{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: Combine the exponents in the denominator using the property of exponents that states when you multiply like bases, you add the exponents. $\newline$ $x^{\frac{5}{3}} \times x^{\frac{1}{3}} = x^{\frac{5}{3} + \frac{1}{3}}$
Add Exponents: Add the exponents in the denominator. $\newline$ $\frac{5}{3} + \frac{1}{3} = \frac{6}{3}$ $\newline$ $x^{\frac{5}{3} + \frac{1}{3}} = x^{\frac{6}{3}}$
Simplify Exponent: Simplify the exponent in the denominator. $\newline$ $\frac{6}{3} = 2$ $\newline$ $x^{\frac{6}{3}} = x^2$
Rewrite Expression: Rewrite the original expression with the simplified denominator. $\newline$ $x^{\frac{2}{3}} / (x^{\frac{5}{3}} \cdot x^{\frac{1}{3}}) = x^{\frac{2}{3}} / x^2$
Apply Property: Apply the property of exponents that states when you divide like bases, you subtract the exponents. $\newline$ $x^{\frac{2}{3}} / x^2 = x^{\frac{2}{3} - 2}$
Convert Exponent: Convert the whole number exponent to a fraction to subtract the exponents. $\newline$ $2 = \frac{6}{3}$ $\newline$ $x^{\frac{2}{3} - 2} = x^{\frac{2}{3} - \frac{6}{3}}$
Subtract Exponents: Subtract the exponents. $\newline$ $\frac{2}{3} - \frac{6}{3} = -\frac{4}{3}$ $\newline$ $x^{\left(\frac{2}{3} - \frac{6}{3}\right)} = x^{-\frac{4}{3}}$
Rewrite Exponent: Since we want all exponents to be positive, we can rewrite $x^{(-4/3)}$ as $1/x^{(4/3)}$ . $\newline$ $x^{(-4/3)} = 1/x^{(4/3)}$