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Simplify. Assume all variables are positive. $\newline$ $z^{\frac{1}{3}} \cdot z^{\frac{5}{3}}$ $\newline$ 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. $\newline$ ______ $\newline$

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Simplify expressions involving rational exponents

Full solution

Q. Simplify. Assume all variables are positive.

\newline

z^{\frac{1}{3}} \cdot z^{\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

______

\newline

Identify Equation: Identify the equation and apply the property of exponents for multiplication. $\newline$ When multiplying powers with the same base, we add the exponents. $\newline$ $z^{\frac{1}{3}} \times z^{\frac{5}{3}} = z^{\frac{1}{3} + \frac{5}{3}}$
Add Exponents: Add the exponents. $\newline$ $\frac{1}{3} + \frac{5}{3} = \frac{6}{3}$
Simplify Fraction: Simplify the fraction $\frac{6}{3}$ . $\frac{6}{3} = 2$
Write Final Answer: Write the final answer using the simplified exponent. $z^{\frac{6}{3}} = z^2$

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(A) The limit exists, and we found it!

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(B) The limit doesn't exist (probably an asymptote).

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(C) The result is indeterminate.

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