Simplify. Assume all variables are positive. $\newline$ $\frac{b^{\frac{8}{3}}}{b^{\frac{4}{3}} \cdot b^{\frac{2}{3}}}$ $\newline$ 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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Algebra 2

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Q. Simplify. Assume all variables are positive.

\newline

\frac{b^{\frac{8}{3}}}{b^{\frac{4}{3}} \cdot b^{\frac{2}{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

______

Simplify Denominator: We have the expression: $b^{\frac{8}{3}} / (b^{\frac{4}{3}} * b^{\frac{2}{3}})$ $\newline$ First, let's simplify the denominator using the property of exponents that states $a^{m} * a^{n} = a^{m+n}$ . $\newline$ $b^{\frac{4}{3}} * b^{\frac{2}{3}} = b^{\frac{4}{3} + \frac{2}{3}}$
Add Exponents: Now, let's add the exponents in the denominator. $\newline$ $\frac{4}{3} + \frac{2}{3} = \frac{6}{3}$ $\newline$ So, $b^{\frac{4}{3}} \times b^{\frac{2}{3}} = b^{\frac{6}{3}}$
Rewrite Denominator: Since $\frac{6}{3}$ simplifies to $2$ , we can rewrite the denominator as $b^2$ . $\newline$ $b^{\frac{6}{3}} = b^2$ $\newline$ Now, our expression is $\frac{b^{\frac{8}{3}}}{b^2}$
Apply Exponent Property: Next, we apply the property of exponents that states $a^{m} / a^{n} = a^{m-n}$ to simplify the expression. $\newline$ $b^{\frac{8}{3}} / b^{2} = b^{\frac{8}{3} - 2}$
Subtract Exponents: We need to express $2$ as a fraction with a denominator of $3$ to subtract it from $\frac{8}{3}$ . $2$ can be written as $\frac{6}{3}$ , so the expression becomes $b^{\left(\frac{8}{3} - \frac{6}{3}\right)}$ .
Final Answer: Now, let's subtract the exponents. $\newline$ $\frac{8}{3} - \frac{6}{3} = \frac{2}{3}$ $\newline$ So, $b^{\frac{8}{3}} / b^2 = b^{\frac{2}{3}}$
Final Answer: Now, let's subtract the exponents. $\newline$ $\frac{8}{3} - \frac{6}{3} = \frac{2}{3}$ $\newline$ So, $b^{\frac{8}{3}} / b^2 = b^{\frac{2}{3}}$ The expression is now fully simplified, and there are no negative exponents. $\newline$ The final answer is $b^{\frac{2}{3}}$ .