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Q81.

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Found in: Page 267

Precalculus Mathematics for Calculus

Book edition 7th Edition
Author(s) James Stewart, Lothar Redlin, Saleem Watson
Pages 948 pages
ISBN 9781337067508

Rationalize Put each fractional expression into standard form by rationalizing the denominator.a. $\frac{\mathbf{1}}{\sqrt{\mathbf{5}\mathbf{x}}}$b. $\sqrt{\frac{\mathbf{x}}{\mathbf{5}}}$c. $\sqrt[\mathbf{5}]{\frac{\mathbf{1}}{{\mathbf{x}}^{\mathbf{3}}}}$

1. The simplified form of $\frac{1}{\sqrt{5x}}$ is $\frac{\sqrt{5x}}{5x}$.
2. The simplified form of $\sqrt{\frac{x}{5}}$ is $\frac{\sqrt{5x}}{5}$.
3. The simplified form of $\sqrt[5]{\frac{1}{{x}^{3}}}$ is $\frac{\sqrt[5]{{x}^{2}}}{x}$.
See the step by step solution

Part a. Step 1. Given information.

The given expression is $\frac{1}{\sqrt{5x}}$.

part a. Step 2. Write the concept.

Rationalizing the denominator is a procedure to eliminate the radical in the denominator by multiplying both numerator and denominator by an appropriate expression.

If the denominator is of the form $\sqrt{a}$, then multiply numerator and denominator by $\sqrt{a}$.

If the denominator is of the form $\sqrt[n]{{a}^{m}},m, then multiply numerator and denominator by $\sqrt[n]{{a}^{n-m}}$.

Part a. Step 3. Determine the simplified form of the expression.

The given expression can be written as:

$\begin{array}{c}\frac{1}{\sqrt{5x}}=\frac{1}{\sqrt{5x}}×\frac{\sqrt{5x}}{\sqrt{5x}}\\ =\frac{\sqrt{5x}}{5x}\end{array}$

Part b. Step 1. Given information.

The given expression is $\sqrt{\frac{x}{5}}$.

Part b. Step 2. Write the concept.

Rationalizing the denominator is a procedure to eliminate the radical in the denominator by multiplying both numerator and denominator by an appropriate expression.

If the denominator is of the form $\sqrt{a}$, then multiply numerator and denominator by $\sqrt{a}$.

If the denominator is of the form $\sqrt[n]{{a}^{m}},m, then multiply numerator and denominator by $\sqrt[n]{{a}^{n-m}}$.

Part c. Step 3. Determine the simplified form of the expression.

The given expression can be written as:

$\begin{array}{c}\sqrt{\frac{x}{5}}=\frac{\sqrt{x}}{\sqrt{5}}\\ =\frac{\sqrt{x}}{\sqrt{5}}×\frac{\sqrt{5}}{\sqrt{5}}\\ =\frac{\sqrt{x×5}}{5}\\ =\frac{\sqrt{5x}}{5}\end{array}$

Part c. Step 1. Given information.

The given expression is $\sqrt[5]{\frac{1}{{x}^{3}}}$.

Part c.  Step 2. Write the concept.

Rationalizing the denominator is a procedure to eliminate the radical in the denominator by multiplying both numerator and denominator by an appropriate expression.

If the denominator is of the form $\sqrt{a}$, then multiply numerator and denominator by $\sqrt{a}$.

If the denominator is of the form $\sqrt[n]{{a}^{m}},m, then multiply numerator and denominator by $\sqrt[n]{{a}^{n-m}}$.

Part c. Step 3. Determine the simplified form of the expression.

The given expression can be written as:

$\begin{array}{c}\sqrt[5]{\frac{1}{{x}^{3}}}=\frac{1}{\sqrt[5]{{x}^{3}}}\\ =\frac{1}{\sqrt[5]{{x}^{3}}}×\frac{\sqrt[5]{{x}^{5-3}}}{\sqrt[5]{{x}^{5-3}}}\\ =\frac{1}{\sqrt[5]{{x}^{3}}}×\frac{\sqrt[5]{{x}^{2}}}{\sqrt[5]{{x}^{2}}}\\ =\frac{\sqrt[5]{{x}^{2}}}{\sqrt[5]{{x}^{3+2}}}\\ =\frac{\sqrt[5]{{x}^{2}}}{\sqrt[5]{{x}^{5}}}\\ =\frac{\sqrt[5]{{x}^{2}}}{x}\end{array}$