/*! 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 97 Solve each equation. $$x^{-3}=... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 97

Solve each equation. $$x^{-3}=8$$

Short Answer

Expert verified
x = \frac{1}{2}.

Step by step solution

01

- Rewrite the equation in terms of positive exponents

Start by rewriting the equation using positive exponents. Since the original equation is \(x^{-3} = 8\), it can be rewritten as \ \ \(\frac{1}{x^3} = 8\).
02

- Invert both sides of the equation

Next, to get rid of the fraction, invert both sides of the equation. This will give you \ \ \(x^3 = \frac{1}{8}\).
03

- Take the cube root of both sides

To solve for \(x\), take the cube root of both sides of the equation. This yields \ \ \(x = \sqrt[3]{\frac{1}{8}}\).
04

- Simplify the cube root

Finally, simplify the cube root on the right-hand side. \ \ \(x = \frac{1}{\sqrt[3]{8}} = \frac{1}{2}\).

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.

Positive Exponents
Understanding exponents is essential in algebra. An exponent tells how many times to multiply a number by itself. Positive exponents are straightforward. For example, in the expression \(x^3\), the exponent 3 means that\(x\) is multiplied by itself three times: \(x \cdot x \cdot x\). These operations are fundamental in solving more complex algebraic equations.

In the exercise, you start with a negative exponent \(x^{-3} = 8\). To make it easier to work with, transform the negative exponent into a positive one.
  • A negative exponent indicates a reciprocal. So, \(x^{-3}\) becomes \(\frac{1}{x^3}\).
Now, the equation is simpler: \(\frac{1}{x^3} = 8\). This transformation sets the stage for solving for \(x\).
Cube Root
A cube root functions as the inverse of cubing a number. When you cube a number, you raise it to the power of three, multiplying it by itself twice more. For instance, \(2^3 = 8\), and the cube root of 8 returns us to 2.

In our exercise, once you have \(x^3 = \frac{1}{8}\), you take the cube root of both sides to solve for \(x\). This action will isolate \(x\): \[x = \sqrt[3]{\frac{1}{8}}\].

Remember:
  • The cube root of a fraction \(\frac{a}{b}\)\

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

Solve each equation. $$3+\sqrt{x^{2}-8 x}=0$$

Simplify. $$(5 a \sqrt{a b})^{2}$$

Use a calculator to find approximate solutions to the following equations. Round your answers to three decimal places. $$(x-1)^{-3 / 4}=7.065$$

Simplify. $$\sqrt[3]{16 a^{4}}+a \sqrt[3]{2 a}$$

Solve each problem by writing an equation and solving it. Find the exact answer and simplify it using the rules for radicals. Find the length of the side of a square whose diagonal is 8 feet.
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Pure Maths

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.