/*! 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 22 Multiply. $$\sqrt[4]{x-1} \sqr... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 22

Multiply. $$\sqrt[4]{x-1} \sqrt[4]{x^{2}+x+1}$$

Short Answer

Expert verified
\( \root[4]{(x-1)} \root[4]{(x^2+x+1)} = \root[4]{x^3 - 1} \)

Step by step solution

01

Understand the problem

The exercise requires the multiplication of two fourth roots: \(\root[4]{x-1}\) and \(\root[4]{x^{2}+x+1}\). The goal is to combine these under a single radical.
02

Use the property of radicals

Recall that for any positive numbers \(a\) and \(b\), \(\root[n]{a} \root[n]{b} = \root[n]{ab}\). Apply this property to combine the two fourth roots.
03

Combine the fourth roots

Using the property from Step 2, combine the expressions under a single fourth root: \[ \root[4]{(x-1)} \root[4]{(x^2+x+1)} = \root[4]{(x-1)(x^2+x+1)} \]
04

Simplify the expression

Simplify the expression inside the radical: \[ (x-1)(x^2+x+1) \]Use the distributive property to multiply: \[ x(x^2+x+1) - 1(x^2+x+1) = x^3 + x^2 + x - x^2 - x - 1 = x^3 - 1 \]
05

Final expression under a single radical

The simplified form of the expression inside the radical is \(x^3 - 1\). Therefore, combining the radicals gives: \[ \root[4]{(x-1)(x^2+x+1)} = \root[4]{x^3 - 1} \]

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.

headline of the respective core concept
Multiplying radicals can seem tricky at first, but it's manageable if you break it down step by step.

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

f(x)\( and \)g(x)\( are as given. Find \)(f \cdot g)(x) \cdot$ Assume that all variables represent non-negative real numbers. $$ f(x)=\sqrt[4]{2 x}+5 \sqrt{2 x}, g(x)=\sqrt[3]{2 x} $$

Consider the function g given by $$g(z)=\frac{z^{4}-z^{2}}{z-1}$$ Find \(g(2-3 i)\)

Find the midpoint of the segment with the given endpoints. $$ (-3.4,8.1) \text { and }(4.8,-8.1) $$

Let \(f(x)=x^{2} .\) Find each of the following. Write the slope-intercept equation of the line that is perpendicular to the line \(y=\frac{1}{2} x-7\) and has a \(y\) -intercept of \((0,12)\)

Perform the indicated operation and simplify. Assume that all variables represent positive real numbers. Write the answer using radical notation. $$ \sqrt[10]{a} \sqrt[5]{a^{2}} $$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

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.