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Simplify. Assume all variables are positive. $\newline$ $\frac{s^{\frac{5}{4}}}{s^{\frac{7}{4}}}$ $\newline$ 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{5}{4}}}{s^{\frac{7}{4}}}

\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: $\newline$ $s^{\frac{5}{4}}/s^{\frac{7}{4}}$ $\newline$ Which operation will be applied with exponent? $\newline$ When dividing powers with the same base, the exponents are subtracted.
Apply Rule for Dividing Exponents: Apply the rule for dividing exponents with the same base: $s^{\frac{5}{4}}/s^{\frac{7}{4}} = s^{\frac{5}{4} - \frac{7}{4}}$ Perform the subtraction of the exponents.
Subtract Exponents: Subtract the exponents: $\newline$ $s^{\frac{5}{4} - \frac{7}{4}} = s^{-\frac{2}{4}}$ $\newline$ Simplify the exponent by reducing the fraction.
Reduce Fraction: Reduce the fraction $-\frac{2}{4}$ to its simplest form: $\newline$ $-\frac{2}{4} = -\frac{1}{2}$ $\newline$ So, $s^{-\frac{2}{4}} = s^{-\frac{1}{2}}$ $\newline$ Since we want the exponent to be positive, we can rewrite the expression using the property of negative exponents.
Rewrite with Positive Exponent: Rewrite the expression with a positive exponent: $\newline$ $s^{(-\frac{1}{2})} = \frac{1}{s^{(\frac{1}{2})}}$ $\newline$ This is the simplified form of the original expression with a positive exponent.

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