Simplify. Assume all variables are positive. $\newline$ $\frac{x^{1/4}}{x^{3/4} \cdot x^{7/4}}$ $\newline$ Write your answer in the form $A$ or $A/B$ , 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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Precalculus

Simplify expressions involving rational exponents

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Q. Simplify. Assume all variables are positive.

\newline

\frac{x^{1/4}}{x^{3/4} \cdot x^{7/4}}

\newline

Write your answer in the form

A

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

______

Combine exponents in denominator: Combine the exponents in the denominator using the property of exponents that states when you multiply like bases, you add the exponents. $\newline$ $x^{\frac{3}{4}} \times x^{\frac{7}{4}} = x^{\frac{3}{4} + \frac{7}{4}}$
Add exponents in denominator: Add the exponents in the denominator. $\newline$ $\frac{3}{4} + \frac{7}{4} = \frac{10}{4}$ $\newline$ So, $x^{\frac{3}{4}} \times x^{\frac{7}{4}} = x^{\frac{10}{4}}$
Simplify exponent in denominator: Simplify the exponent in the denominator. $\newline$ $\frac{10}{4}$ can be simplified to $\frac{5}{2}$ . $\newline$ So, $x^{\frac{3}{4}} \times x^{\frac{7}{4}} = x^{\frac{5}{2}}$
Rewrite with simplified denominator: Rewrite the original expression with the simplified denominator. $\newline$ $x^{\frac{1}{4}} / (x^{\frac{3}{4}} \cdot x^{\frac{7}{4}}) = x^{\frac{1}{4}} / x^{\frac{5}{2}}$
Subtract exponents to divide: Subtract the exponents to divide the expressions with the same base using the property of exponents that states when you divide like bases, you subtract the exponents. $\newline$ $x^{\frac{1}{4}} / x^{\frac{5}{2}} = x^{\frac{1}{4} - \frac{5}{2}}$
Convert exponents to common denominator: Convert the exponents to have a common denominator before subtracting. $\newline$ $\frac{5}{2}$ is the same as $\frac{10}{4}$ , so we have: $\newline$ $x^{\frac{1}{4} - \frac{10}{4}}$
Subtract exponents: Subtract the exponents. $\newline$ $\frac{1}{4} - \frac{10}{4} = -\frac{9}{4}$ $\newline$ So, $x^{\frac{1}{4}} / x^{\frac{5}{2}} = x^{-\frac{9}{4}}$
Rewrite as positive exponent: Since we want the exponent to be positive, we can rewrite $x^{(-9/4)}$ as $1 / x^{(9/4)}$ .