/*! 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 Simplify. $$\left(\frac{5 a}{b... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 48

Simplify. $$\left(\frac{5 a}{b}\right)^{2}$$

Short Answer

Expert verified
\( \left(\frac{5 a}{b}\right)^{2} = \frac{25a^2}{b^2} \)

Step by step solution

01

Understand the Expression

The given expression is \( \left(\frac{5a}{b}\right)^2 \). This is a fraction raised to the power of 2, meaning each part of the fraction must be squared separately.
02

Square the Numerator

The numerator of the fraction, \(5a\), should be squared. This means \((5a)^2 = 5^2 \cdot a^2 = 25a^2\).
03

Square the Denominator

The denominator of the fraction, \(b\), is squared next. This gives us \(b^2\).
04

Combine the Results

Place the squared numerator and denominator back in fraction form. The simplified expression becomes \( \frac{25a^2}{b^2} \).

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.

Simplifying Algebraic Expressions
Simplifying algebraic expressions helps to make them more manageable by comparing and manipulating like terms. When we simplify, we aim to express an equation or expression in the most straightforward form possible. In the exercise, we were given \[\left(\frac{5a}{b}\right)^2\]meaning it must be simplified to a clearer, simpler expression while preserving its value.
  • Identify powers and perform necessary operations.
  • Combine like terms, if applicable.
  • Maintain equal operation order to ensure a mathematically sound process.
Simplifying makes it easier to understand and useful for subsequent algebraic operations or solutions.
Squaring a Fraction
Squaring a fraction involves multiplying the fraction by itself. This means each term in the fraction is squared. Given the fraction\(\frac{5a}{b}\)to be squared as \[\left(\frac{5a}{b}\right)^2\], it implies:
  • Square the numerator \((5a)\), resulting in \( (5a)^2 = 25a^2\).
  • Square the denominator \(b\), leading to \(b^2\).
The result is a new fraction \[\frac{25a^2}{b^2}\].
This operation allows you to distribute the exponent to both the numerator and the denominator, maintaining the equality of the expression.
Numerator and Denominator Operations
When handling fractions, performing operations on the numerator and denominator is essential. These operations ensure the fraction is in its simplest or required form. Here鈥檚 a closer look at how these operations work:
  • Numerator Operations: Any operation or exponent applied to the numerator directly impacts the fraction's overall value. In our exercise, squaring the numerator \(5a\) directly led to the simplification \(25a^2\).
  • Denominator Operations: Changes to the denominator affect how the entire fraction scales. Squaring the denominator \(b\) resulted in \(b^2\), maintaining balance in the fraction.
Understanding these operations makes the process of working with more complex fractions straightforward. You can more confidently tackle equations, knowing how changes to either the numerator or denominator factor into the entire fraction.

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

Determine whether each statement is true or false . If false, give a counter example. Two spheres can intersect in one point.

Write an equation for each circle. diameter with endpoints at \((5,-4)\) and \((-1,6)\)

Consider a rectangular prism with the given coordinates. Find the coordinates of the vertices of the prism after the translation. $$\begin{aligned}&P(-2,-3,3), Q(-2,0,3), R(0,0,3), S(0,-3,3), T(-2,0,0), U(-2,-3,0)\\\&V(0,-3,0), \text { and } W(0,0,0) ;(x, y, z) \rightarrow(x+2, y+5, z-5)\end{aligned}$$

Find the surface area of each cone. Round to the nearest tenth. diameter \(=16 \mathrm{cm},\) slant height \(=13.5 \mathrm{cm}\)

Find the surface area of each cone. Round to the nearest tenth. \(r=7\) meters, \(h=10\) meters
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

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.