Simplify. Assume all variables are positive. $\newline$ $\frac{x^{1/4}}{x^{3/4} \cdot x^{9/4}}$ $\newline$ 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^{9/4}}

\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. $\newline$ $x^{\frac{3}{4}} \times x^{\frac{9}{4}} = x^{\frac{3}{4} + \frac{9}{4}}$
Add exponents in denominator: Add the exponents in the denominator. $\newline$ $\frac{3}{4} + \frac{9}{4} = \frac{12}{4}$ $\newline$ $x^{\frac{3}{4}} \times x^{\frac{9}{4}} = x^{\frac{12}{4}}$
Simplify exponent in denominator: Simplify the exponent in the denominator. $\frac{12}{4} = 3$ $x^{\frac{12}{4}} = x^3$
Rewrite with simplified denominator: Rewrite the original expression with the simplified denominator. $\frac{x^{\frac{1}{4}}}{x^3}$
Apply property of exponents: Apply the property of exponents that states when you divide like bases, you subtract the exponents. $\newline$ $x^{\frac{1}{4}} / x^3 = x^{\frac{1}{4} - 3}$
Convert whole number exponent: Convert the whole number exponent to a fraction to have a common denominator. $\newline$ $3 = \frac{12}{4}$ $\newline$ $x^{\frac{1}{4} - 3} = x^{\frac{1}{4} - \frac{12}{4}}$
Subtract exponents: Subtract the exponents. $\newline$ $\frac{1}{4} - \frac{12}{4} = -\frac{11}{4}$ $\newline$ $x^{\frac{1}{4} - \frac{12}{4}} = x^{-\frac{11}{4}}$
Rewrite with positive exponent: Since we want the exponent to be positive, we can rewrite the expression with a positive exponent by taking the reciprocal of the base. $\newline$ $x^{(-\frac{11}{4})} = \frac{1}{x^{\frac{11}{4}}}$