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Intermediate Algebra

OPENSTAX

$Math Studyset 91影视 Explanations$ Math

OER 2017 Edition 路 1346 pages

ISBN: 9780998625720

Chapter 8: Q. 8.14 (page 750)

Simplify: $(a) \sqrt{y^{2}} (b) \sqrt[3]{p^{3}} (c) \sqrt[4]{z^{4}} (d) \sqrt[5]{q^{5}}$

Short Answer

Expert verified

The solutions of (a), (b), (c) and (d) are $|y|, p, |z| a n d q$ respectively.

Step by step solution

01

Part (a) Step 1. Given information

The root: $\sqrt{y^{2}}$

02

Part (a) Step 2. Calculation

We use the absolute value to be sure to get the positive root.

$\sqrt{y^{2}} = |y|$

Since the index n is even, $\sqrt[2]{a^{n}} = |a|$

Therefore, the solution is $|y|$

03

Part (b) Step 1. Given information

The root: $\sqrt[3]{p^{3}}$

04

Part (b) Step 2. Calculation

This is an odd indexed root so there is no need for an absolute value sign.

$\sqrt[3]{p^{3}} = p$

Since the index n is odd, $\sqrt[n]{a^{n}} = a$

Therefore, the solution is $p$

05

Part (c) Step 1. Given information

The root: $\sqrt[4]{z^{4}}$

06

Part (c) Step 2. Calculation

We use the absolute value to be sure to get the positive root.

$\sqrt[4]{z^{4}} = |z|$

Since the index n is even, $\sqrt[n]{a^{n}} = |a|$

The solution is $|z|$

07

Part (d) Step 1. Given information

The root: $\sqrt[5]{q^{5}}$

08

Part (d) Step 2. Calculation

This is an odd indexed root so there is no need for an absolute value sign.

$\sqrt[5]{q^{5}} = q$

Since the index n is odd, $\sqrt[n]{a^{n}} = a$

Therefore, the solution is $q$

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Most popular questions from this chapter

In the following exercises, solve.
$\sqrt[3]{4 x - 1} = 3$

In the following exercises, simplify using absolute value signs as needed. $\sqrt{175 n^{13}}, \sqrt[5]{512 p^{5}}, \sqrt[4]{324 q^{7}}$

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(a) $\sqrt{\frac{32 x^{5} y^{3}}{18 x^{3} y}}$

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(a) $- 64^{\frac{3}{2}}$

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In the following exercises, solve.
$\sqrt{u - 3} - 3 = u$

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