Simplify. Assume all variables are positive. $\newline$ $s^{\frac{1}{3}} \cdot s^{\frac{8}{3}}$ $\newline$ 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

s^{\frac{1}{3}} \cdot s^{\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 be simplified. $\newline$ We have the expression $s^{\frac{1}{3}} \cdot s^{\frac{8}{3}}$ and we need to simplify it.
Apply Product Rule: Apply the product rule for exponents. $\newline$ The product rule states that when multiplying two powers with the same base, you add the exponents. So, we add the exponents $\frac{1}{3}$ and $\frac{8}{3}$ . $\newline$ $s^{\frac{1}{3}} \times s^{\frac{8}{3}} = s^{\frac{1}{3} + \frac{8}{3}}$
Perform Exponent Addition: Perform the addition of the exponents. $\newline$ Add the fractions $\frac{1}{3}$ and $\frac{8}{3}$ . $\newline$ $\frac{1}{3} + \frac{8}{3} = \frac{(1 + 8)}{3}$ $\newline$ $\frac{1}{3} + \frac{8}{3} = \frac{9}{3}$
Simplify Fraction: Simplify the fraction $9/3$ . The fraction $9/3$ simplifies to $3$ . $9/3 = 3$
Write Final Expression: Write the final simplified expression. $\newline$ Since $\frac{9}{3}$ simplifies to $3$ , we have: $\newline$ $s^{\frac{1}{3} + \frac{8}{3}} = s^3$