Simplify. Assume all variables are positive. $\newline$ $\frac{x^{\frac{2}{3}}}{x^{\frac{8}{3}}}$ $\newline$ 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{8}{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

______

Identify expression: Identify the expression to simplify. $\newline$ We have the expression $x^{\frac{2}{3}}/x^{\frac{8}{3}}$ and we need to simplify it.
Apply quotient rule: Apply the quotient rule for exponents. $\newline$ The quotient rule states that when we divide two exponents with the same base, we subtract the exponents: $a^m / a^n = a^{(m-n)}$ . $\newline$ So, $x^{\frac{2}{3}}/x^{\frac{8}{3}} = x^{(\frac{2}{3}-\frac{8}{3})}$ .
Perform subtraction: Perform the subtraction of the exponents. $(\frac{2}{3}) - (\frac{8}{3}) = -\frac{6}{3}$ .
Simplify result: Simplify the result of the subtraction. $-\frac{6}{3}$ simplifies to $-2$ .
Write simplified exponent: Write the simplified exponent with the base $x$ . $x^{(\frac{2}{3})-(\frac{8}{3})} = x^{-2}$ .
Use property: Since we want all exponents to be positive, we use the property that $x^{-a} = \frac{1}{x^a}$ . $\newline$ Therefore, $x^{-2} = \frac{1}{x^2}$ .