/*! 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 31 $$\left(\frac{w}{2}\right)^{3}$$... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 31

$$\left(\frac{w}{2}\right)^{3}$$

Short Answer

Expert verified
\( \frac{w^3}{8} \)

Step by step solution

01

Understand the expression

The given expression is \( \left( \frac{w}{2} \right)^3 \), which means raising the fraction \( \frac{w}{2} \) to the power of 3.
02

Apply the exponent to both numerator and denominator

When raising a fraction to a power, apply the power to both the numerator and the denominator separately. This gives us: \( \left( \frac{w}{2} \right)^3 = \frac{w^3}{2^3} \).
03

Simplify the denominator

Calculate the denominator by raising 2 to the power of 3: \( 2^3 = 8 \). So the expression becomes: \( \frac{w^3}{8} \).

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.

Exponent Rules
Let's start by understanding some basic rules of exponents. An exponent tells us how many times to multiply the base by itself. For example, in the expression \(2^3\), the base is 2, and the exponent is 3, which means we multiply 2 by itself 3 times: \(2 \times 2 \times 2 = 8\).
  • When you have an expression like \(a^m \times a^n\), you add the exponents: \(a^{m+n}\).
  • If you have \((a^m)^n\), you multiply the exponents: \(a^{m \times n}\).
  • For division, \(\frac{a^m}{a^n} = a^{m-n}\).
  • For a fraction raised to an exponent, apply the exponent to both the numerator and denominator: \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\).

These rules simplify calculations involving exponents and make it easier to handle expressions with powers.
Numerator and Denominator
Fractions are composed of two parts: the numerator and the denominator. The numerator is the top number, representing how many parts we have. The denominator is the bottom number, showing how many equal parts the whole is divided into.
  • For \(\frac{3}{4}\), 3 is the numerator, and 4 is the denominator.
  • If you raise a fraction to a power, you apply the exponent to both parts separately. For example, \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\). In this exercise, our fraction is \(\frac{w}{2}\).

The exponent of 3 is applied to both the numerator \(w\) and the denominator \(2\), resulting in \(\frac{w^3}{2^3}\). This breaks the problem into more manageable parts, making it easier to simplify.
Simplification
The final step in solving our expression is simplification.
  • We start with \(\left(\frac{w}{2}\right)^3 = \frac{w^3}{2^3}\).
  • Next, we simplify the denominator. \(2^3\) means multiplying 2 by itself three times: \(2 \times 2 \times 2 = 8\).

So, our expression simplifies from \(\frac{w^3}{2^3}\) to \(\frac{w^3}{8}\).

Simplifying expressions is crucial because it helps find the simplest form of an answer, making it easier to interpret and apply in further calculations or real-world scenarios.

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

Factor each polynomial completely. $$ -2 w^{2}+18 w+20 $$

Factor each polynomial by grouping. $$ 2 m w-5 w-2 n+5 $$

Solve each equation. $$(x-1)(x+3)(x-9)=0$$

Which of the following is not a difference of two squares? Explain. a) \(16 a^{8} y^{4}-25 c^{12}\) b) \(a^{9}-b^{4}\) c) \(t^{90}-1\) d) \(x^{2}-196\)

Factor each polynomial using the trial-and-error method. $$ z^{2}-2 z-35 $$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Calculus

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.