/*! 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 25 Multiply. $$\sqrt[7]{\frac{x-3... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 25

Multiply. $$\sqrt[7]{\frac{x-3}{4}} \sqrt[7]{\frac{5}{x+2}}$$

Short Answer

Expert verified
\(\frac{\root{7}{(x-3) \times 5}}{\root{7}{4(x+2)}}\)

Step by step solution

01

Understand the problem

You need to multiply \(\frac{x-3}{4}\frac{5}{x+2}\) under a single radical expression of index 7.
02

Multiply the radicands

Under the radicals, multiply the expressions: \(\frac{x-3}{4} \times \frac{5}{x+2}\).
03

Simplify the expression inside the radical

Simplify the multiplication: \(\frac{(x-3) \times 5}{4 \times (x+2)}\).
04

Combine under single radical

Now write the simplified expression under a single radical: \(\frac{(x-3)5}{4(x+2)}\).

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.

Understanding Radicals
Radicals are expressions that involve roots, such as square roots, cube roots, and in this case, seventh roots. Specifically, if you see something like \(\frac{a}{b}\)\(\sqrt[7]{\frac{a}{b}}\), it stands for the seventh root of the fraction \(\frac{a}{b}\). Radicals are essential in various areas of math and often appear in equations and algebraic expressions. When dealing with radicals, we must understand how to manipulate these roots correctly, particularly when multiplying or simplifying them.
Algebraic Multiplication
Algebraic multiplication is the process of multiplying two algebraic expressions. When you multiply expressions involving radicals, you follow the basic multiplication rules. For example, multiplying \(\frac{x-3}{4}\) by \(\frac{5}{x+2}\) involves multiplying the numerators together and the denominators together. This results in \(\frac{(x-3) \times 5}{4 \times (x+2)}\). This concept is essential when working with more complex expressions, as it allows you to simplify and combine them efficiently.
The Simplification Process
Simplification means breaking down a complex expression into a simpler form. For this exercise, start by multiplying the numerators and denominators: \(\frac{(x-3) \times 5}{4 \times (x+2)}\). After performing this multiplication, we get \(\frac{5(x-3)}{4(x+2)}\). The goal of simplification is to make the expression as straightforward as possible, which often involves factoring, canceling out common terms, or combining like terms. By simplifying under a single radical, we get an easier-to-understand expression: \(\frac{5(x-3)}{4(x+2)}\).
Rational Expressions
A rational expression is a fraction where the numerator and the denominator are both polynomials. In this exercise, \(\frac{x-3}{4}\) and \(\frac{5}{x+2}\) are rational expressions. To multiply them under a radical of index 7, you combine them, which results in another rational expression: \(\frac{5(x-3)}{4(x+2)}\). Rational expressions are significant in algebra because they can be manipulated and simplified similarly to numerical fractions, allowing for the solving and graphing of more complex equations. Understanding rational expressions make it easier to work with various algebraic problems and solutions.

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

Contracting. Cleary Construction has a hose attached to their insulation blower that permits them to reach anywhere in a circular area of \(6160 \mathrm{ft}^{2} .\) Find the dimensions of the largest square room with 12 -ft ceilings in which they could reach all corners with the hose while leaving the blower centrally located. Assume that the blower sits on the floor.

Perform the indicated operation and simplify. Assume that all variables represent positive real numbers. Write the answer using radical notation. $$ \sqrt{x y^{3}} \sqrt[3]{x^{2} y} $$

Find the midpoint of the segment with the given endpoints. $$ (4.1,6.9) \text { and }(5.2,-8.9) $$

f(x)\( and \)g(x)\( are as given. Find \)(f \cdot g)(x) \cdot$ Assume that all variables represent non-negative real numbers. $$ f(x)=x+\sqrt{7}, g(x)=x-\sqrt{7} $$

f(x)\( and \)g(x)\( are as given. Find \)(f \cdot g)(x) \cdot$ Assume that all variables represent non-negative real numbers. $$ f(x)=x-\sqrt{2}, g(x)=x+\sqrt{6} $$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

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.