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Q81.

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Precalculus Mathematics for Calculus

Found in: Page 267

Book edition 7th Edition

Author(s) James Stewart, Lothar Redlin, Saleem Watson

Pages 948 pages

ISBN 9781337067508

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Short Answer

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

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

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Part a. Step 1. Given information.

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

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 $\sqrt[n]{a^{m}}, m < n$ , 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:

$\begin{matrix} \frac{1}{\sqrt{5 x}} = \frac{1}{\sqrt{5 x}} \times \frac{\sqrt{5 x}}{\sqrt{5 x}} \\ = \frac{\sqrt{5 x}}{5 x} \end{matrix}$

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 $\sqrt[n]{a^{m}}, m < n$ , 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:

$\begin{matrix} \sqrt{\frac{x}{5}} = \frac{\sqrt{x}}{\sqrt{5}} \\ = \frac{\sqrt{x}}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} \\ = \frac{\sqrt{x \times 5}}{5} \\ = \frac{\sqrt{5 x}}{5} \end{matrix}$

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 $\sqrt[n]{a^{m}}, m < n$ , 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:

$\begin{matrix} \sqrt[5]{\frac{1}{x^{3}}} = \frac{1}{\sqrt[5]{x^{3}}} \\ = \frac{1}{\sqrt[5]{x^{3}}} \times \frac{\sqrt[5]{x^{5 - 3}}}{\sqrt[5]{x^{5 - 3}}} \\ = \frac{1}{\sqrt[5]{x^{3}}} \times \frac{\sqrt[5]{x^{2}}}{\sqrt[5]{x^{2}}} \\ = \frac{\sqrt[5]{x^{2}}}{\sqrt[5]{x^{3 + 2}}} \\ = \frac{\sqrt[5]{x^{2}}}{\sqrt[5]{x^{5}}} \\ = \frac{\sqrt[5]{x^{2}}}{x} \end{matrix}$

Most popular questions for Math Textbooks

Intervals Express the interval in terms of inequalities, and then graph the interval $[2, 8)$ .

Find the indicated set if Find the indicated set if

$\begin{array}{l} A = {x | x \geq - 2} \\ B = {x | x < 4} \\ C = {x | - 1 < x \leq 5} \end{array}$

a . A \cap C b . A \cap B

Inequalities Place the correct symbol $(<, >, or =)$ in the space.

a. $3___\frac{7}{2}$

b. $- 3___- \frac{7}{2}$

c. $3.5___\frac{7}{2}$

Rational Exponents Evaluate each expression.

a. $16^{\frac{1}{4}}$

b. role="math" localid="1648443341033" $- 8^{\frac{1}{3}}$

c. $9^{- \frac{1}{2}}$

Radicals and Exponents Evaluate each expression.

a. $- 2^{3} \cdot {(- 2)}^{0}$

b. $- 2^{- 3} \cdot {(- 2)}^{0}$

c. ${(\frac{- 3}{5})}^{- 3}$

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