Simplify. Assume all variables are positive. $\newline$ $\frac{b^{\frac{2}{3}}}{b^{\frac{4}{3}}}$ $\newline$ 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{b^{\frac{2}{3}}}{b^{\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

______

Identify Equation & Apply Quotient Rule: Identify the equation and apply the quotient rule for exponents. $\newline$ The quotient rule states that when dividing like bases with exponents, you subtract the exponents: $a^m / a^n = a^{m-n}$ . $\newline$ So, $b^{2/3} / b^{4/3} = b^{2/3 - 4/3}$ .
Subtract Exponents: Subtract the exponents. $\newline$ $\frac{2}{3} - \frac{4}{3} = -\frac{2}{3}$ . $\newline$ So, $b^{\frac{2}{3}} / b^{\frac{4}{3}} = b^{-\frac{2}{3}}$ .
Rewrite Negative Exponent: Since we want all exponents to be positive, we can rewrite $b^{-2/3}$ as $1/b^{2/3}$ . This is because a negative exponent indicates the reciprocal of the base raised to the positive of that exponent.
Check Final Answer: Check the final answer for any common variables in the numerator and denominator and ensure all exponents are positive. $\newline$ We have $\frac{1}{b^{\frac{2}{3}}}$ , which has no variables in common between the numerator and denominator, and the exponent is positive.