/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 467 In the following exercises, simp... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 467

In the following exercises, simplify. (a) \(\sqrt[3]{625}\) (b) \(\sqrt[6]{128}\)

Short Answer

Expert verified
(a) \(5^{4/3}\); (b) \(2^{7/6}\)

Step by step solution

01

Understanding Cube Root

To simplify \(\sqrt[3]{625}\), we need to find a number which, when cubed, gives 625.
02

Factorize the Number

625 can be factorized as follows: \(625 = 5\times 5\times 5\times 5 = 5^4\).
03

Simplify Cube Root

The cube root of \(5^3\) is clearly 5, since \(5^3 = 125\). But we need to cube root 625. Since \(625 = 5^4\), we identify that \(\sqrt[3]{5^4} = 5^{4/3}\).
04

Simplified Answer for (a)

Therefore, \(\sqrt[3]{625} = 5^{4/3} \). This is the simplest form we will have without a single integer value.
05

Understanding Sixth Root

To simplify \(\sqrt[6]{128}\), we need to find a number which, when raised to the power of 6, gives 128.
06

Factorize the Number

128 can be factorized as follows: \(128 = 2\times 2\times 2\times 2\times 2\times 2\times 2 = 2^7\).
07

Simplify Sixth Root

The sixth root of \(2^6\) is clearly 2, since \(2^6 = 64\). But we need to sixth root 128. Since \(128 = 2^7\), we identify that \(\sqrt[6]{2^7} = 2^{7/6}\).
08

Simplified Answer for (b)

Therefore, \(\sqrt[6]{128} = 2^{7/6} \). This is the simplest form we will have without a single integer value.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cube Roots
Cube roots help us find a number which, when raised to the power of three (cubed), equals the given number. Think of it as the opposite of cubing a number. When simplifying cube roots, the first step is to factorize the given number.
For example, \( \sqrt[3]{625} \) can be factorized as follows:\

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

In the following exercises, solve. \(\sqrt{2 m+1}+4=0\)

In the following exercises, simplify by rationalizing the denominator. $$ \frac{\sqrt{3}}{\sqrt{m}-\sqrt{5}} $$

In the following exercises, simplify by rationalizing the denominator. $$ \frac{\sqrt{80 p^{3} q}}{\sqrt{5 p q^{5}}} $$

In the following exercises, simplify and rationalize the denominator. $$ -\frac{8}{3 \sqrt{6}} $$

In the following exercises, simplify by rationalizing the denominator. $$ \frac{3}{5 \sqrt{8}} $$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.