/*! 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 29 Express using positive exponents... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 29

Express using positive exponents and, if possible, simplify. $$\left(\frac{a}{2}\right)^{-3}$$

Short Answer

Expert verified
\(\frac{8}{a^3}\).

Step by step solution

01

Understand the problem

The expression is \(\bigg(\frac{a}{2}\bigg)^{-3}\). The goal is to simplify this expression and express it using positive exponents.
02

Apply the negative exponent rule

Recall that \({x^{-n} = \frac{1}{x^n}}\). Applying this rule, we get \(\bigg(\frac{a}{2}\bigg)^{-3} = \frac{1}{\bigg(\frac{a}{2}\bigg)^{3}}\).
03

Simplify the denominator

Next, simplify the denominator \(\bigg(\frac{a}{2}\bigg)^{3}\). This follows the rule \(\big(\frac{x}{y}\big)^n = \frac{x^n}{y^n}\), so \(\bigg(\frac{a}{2}\bigg)^{3} = \frac{a^{3}}{2^{3}}\bigg.\), which simplifies to \(\frac{a^3}{8}\).
04

Combine the fractions

Our expression is now \(\frac{1}{\bigg(\frac{a^3}{8}}\bigg)\). To simplify, multiply by the reciprocal: \(\frac{1}{\frac{a^3}{8}} = \frac{8}{a^3}\).

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.

negative exponent rule
When you encounter a negative exponent in an expression, it might seem confusing at first. The negative exponent rule helps us understand this concept easily. According to the rule, any base with a negative exponent can be rewritten as the reciprocal of the base with a positive exponent. For instance, \(x^{-n} = \frac{1}{x^n}\). This rule transforms an expression with a negative exponent into a fraction, making it simpler to work with.
In our exercise, we applied this rule to \(\left(\frac{a}{2}\right)^{-3}\), converting it to \(\frac{1}{\left(\frac{a}{2}\right)^3}\). This step is crucial as it helps us move forward with positive exponents, making further simplification easier.
fraction exponent rule
Once the negative exponent is handled, the fraction exponent rule is used to simplify further. This rule states that raising a fraction to a power involves raising both the numerator and the denominator to that power separately. Mathematically, it can be represented as \(\left(\frac{x}{y}\right)^n = \frac{x^n}{y^n}\).
In the given problem, we raised the fraction \(\frac{a}{2}\) to the power of 3, resulting in \(\left(\frac{a}{2}\right)^3 = \frac{a^3}{2^3}\). This gives \(\frac{a^3}{8}\) as our new expression. Simplifying each part of the fraction separately ensures that the expression is easy to manage and further simplifies our work.
reciprocal
A reciprocal is a simple yet powerful concept that's widely used in algebra. The reciprocal of a number is simply 1 divided by that number. For instance, the reciprocal of \(x\) is \(\frac{1}{x}\), and the reciprocal of \(\frac{x}{y}\) is \(\frac{y}{x}\). It's like flipping the numerator and the denominator.
In our exercise, after forming the fraction \(\frac{1}{\left(\frac{a^3}{8}\right)}\), we used the reciprocal to simplify further. Flipping the fraction \(\frac{a^3}{8}\), we get \(\frac{8}{a^3}\). Recognizing and applying the reciprocal is vital in these problems as it often leads to the solution in its simplest form.

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

For each pair of functions \(f\) and \(g\), determine the domain of \(f / g\) $$ \begin{aligned} &f(x)=5+x\\\ &g(x)=6-2 x \end{aligned} $$

Replace \(\square\) with \(>,<,\) or \(=\) to write a true sentence. $$ 4^{2} \square 4^{3} $$

To prepare for Section \(5.2,\) review operations with integers (Sections \(1.5-1.7)\) Perform the indicated operations. $$ -3(5) $$

A polynomial in \(x\) has degree \(3 .\) The coefficient of \(x^{2}\) is 3 less than the coefficient of \(x^{3} .\) The coefficient of \(x\) is three times the coefficient of \(x^{2} .\) The remaining constant is 2 more than the coefficient of \(x^{3} .\) The sum of the coefficients is \(-4 .\) Find the polynomial.

F(x)\( and \)g(x)\( are as given. Find a simplified expression for \)F(x)\( if \)F(x)=(f / g)(x)$. $$ f(x)=8 x^{3}+27, g(x)=2 x+3 $$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

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.