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Simplify. Assume all variables are positive. $\newline$ $\frac{s^{\frac{4}{3}}}{s^{\frac{8}{3}}}$ $\newline$ 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. $\newline$ ______

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Simplify expressions involving rational exponents

Full solution

Q. Simplify. Assume all variables are positive.

\newline

\frac{s^{\frac{4}{3}}}{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

______

Identify operation: We have the expression: $\newline$ $s^{\frac{4}{3}} / s^{\frac{8}{3}}$ $\newline$ Which operation will be applied with the exponents? $\newline$ When dividing powers with the same base, the exponents are subtracted.
Apply exponent rule: Apply the exponent rule for division: $s^{\frac{4}{3}} / s^{\frac{8}{3}} = s^{\frac{4}{3} - \frac{8}{3}}$ Perform the subtraction of the exponents.
Calculate difference: Calculate the difference of the exponents: $\newline$ $\frac{4}{3} - \frac{8}{3} = -\frac{4}{3}$ $\newline$ Since we assume all variables are positive and all exponents in the answer should be positive, we need to express the negative exponent as a positive one.
Convert to positive exponent: Convert the negative exponent to a positive exponent by taking the reciprocal of the base: $\newline$ $s^{(-4/3)} = 1/s^{(4/3)}$ $\newline$ This gives us the simplified form with a positive exponent.

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