/*! 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 48 Solve each equation in by making... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 48

Solve each equation in by making an appropriate substitution. $$ 2 x^{2 / 3}+7 x^{1 / 3}-15=0 $$

Short Answer

Expert verified
The solutions to given equation are \(x = \frac{27}{8}\) and \(x = -125\)

Step by step solution

01

Variable Substitution

First, identify the smallest power of \(x\), which in our case is \(\frac{1}{3}\). Now, let's set a new variable such that \(u = x^{1 / 3}\). The equation can then be rewritten in terms of \(u\) as \(2u^2 + 7u - 15 = 0 \)
02

Solving the Equation

This equation is in quadratic form. It can be solved by factorizing or by using the quadratic formula. Here, \(2u^2 + 7u - 15\) can be factored as \((2u - 3)(u + 5) = 0\). By the 'zero-product property', if a product of two factors equals zero, then at least one of the factors must be equal to zero. Therefore, we get two possible solutions for \(u\): \(u = \frac{3}{2}\) or \(u = -5\)
03

Replacing the substituted variable

Now we replace \(u\) with \(x^{1 / 3}\) to get the solutions for original variable \(x\). Hence \(x^{1 / 3} = \frac{3}{2}\) or \(x^{1 / 3} = -5\). Cubing both sides for each of these solutions provides us: \(x = (\frac{3}{2})^3 = \frac{27}{8}\) or \(x = -5^3 = -125\)

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!

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

Use the position formula $$ s=-16 t^{2}+v_{0} t+s_{0} $$ \(\left(v_{0}=\text { initial velocity, } s_{0}=\text { initial position, } t=\text { time }\right)\) to answer Exercises \(49-52 .\) If necessary, round answers to the nearest hundredth of a second. A diver leaps into the air at 20 feet per second from a diving board that is 10 feet above the water. For how many scconds is the diver at least 12 feet above the water?

In Exercises \(29-44,\) perform the indicated operations and write the result in standard form. $$(-2+\sqrt{-11})^{2}$$

Solve each quadratic inequality in Exercises \(1-28\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ -x^{2}+x=0 $$

In Exercises \(1-8,\) add or subtract as indicated and write the result in standard form. $$(7+2 i)+(1-4 i)$$

Solve each rational inequality in Exercises \(29-48,\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ \frac{x^{2}-x-2}{x^{2}-4 x+3}>0 $$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

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.