/*! 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 82 Reduce the given fraction to low... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 82

Reduce the given fraction to lowest terms. $$\frac{96 x^{2}}{14 x^{4}}$$

Short Answer

Expert verified
The fraction \( \frac{96 x^2}{14 x^4} \) reduces to \( \frac{48}{7x^2} \).

Step by step solution

01

Identify the common factors

To reduce the fraction \( \frac{96x^2}{14x^4} \) to lowest terms, first identify the common factors in the numerator and the denominator. Here, both 96 and 14 have the common factor of 2.
02

Simplify the coefficients

Divide the coefficients in the numerator and denominator by their greatest common divisor, which is 2. \( \frac{96}{2} = 48 \) and \( \frac{14}{2} = 7 \). So, the fraction becomes \( \frac{48x^2}{7x^4} \).
03

Simplify the variable expression

Next, simplify the variables by reducing the powers of \( x \). Divide both the numerator and denominator by \( x^2 \). The result becomes \( \frac{x^{2-2}}{x^{4-2}} \), or \( \frac{1}{x^2} \).
04

Assemble the final reduced fraction

Now, combine the simplified coefficient and the simplified variable expression. The fraction \( \frac{48}{7} \) multiplied by \( \frac{1}{x^2} \) simplifies to \( \frac{48}{7x^2} \).

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.

Greatest Common Divisor (GCD)
When simplifying fractions, one of the first steps is to find the greatest common divisor (GCD) of the coefficients. The GCD is the largest number that can divide both numbers without leaving a remainder.
For example, in the original exercise, we had to simplify the fraction \( \frac{96x^2}{14x^4} \). To do this, we found the GCD of 96 and 14. Let's look briefly at how to determine the GCD:
  • List the factors of each number:
    • Factors of 96: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96
    • Factors of 14: 1, 2, 7, 14
  • Identify the largest common factor on both lists, which is 2 in this case.
By dividing the coefficients by the GCD, the numbers become smaller, making it easier to work with the fraction during further simplification steps.
Coefficients
Coefficients are the numerical part of terms in an expression, separate from any variables.
In fractions, they are often at the forefront of simplification. In our problem, after identifying the GCD, we simplified the coefficients of the fraction. Here鈥檚 how it worked in our example:
  • Initially, the coefficients were 96 and 14.
  • By dividing both numbers by 2 (the GCD), we simplified this to 48 and 7.
  • This step ensures that the fraction remains equivalent but now is in a form that can be more easily dealt with.
Simplifying coefficients is a crucial step in the process of reducing fractions and makes further steps clearer and easier.
Variable Expressions
Variable expressions involve terms that include variables raised to different powers. Simplifying them often involves reducing these powers.
In our fraction \( \frac{96x^2}{14x^4} \), the variables \( x \) had different powers in the numerator and denominator. Here's how we approached the simplification:
  • The fraction had \( x^2 \) in the numerator and \( x^4 \) in the denominator.
  • We divided both numerator and denominator by \( x^2 \).
  • This eliminated \( x \) from the numerator (\( x^{2-2} = x^0 = 1 \)) and reduced the denominator to \( x^2 \).
These steps are critical in algebra because simplifying variable expressions allows for clearer understanding and manipulation of more complex equations.

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 the equation and simplify your answer. $$\frac{3}{8} x=\frac{8}{7}$$

Simplify the complex rational expression. $$\frac{\frac{8}{9}+\frac{3}{4}}{-\frac{2}{3}-\frac{1}{6}}$$

Solve the equation and simplify your answer. $$\frac{1}{6} x=\frac{2}{9}$$

The area of a triangle is 51 square inches. If the length of the base is \(8 \frac{1}{2}\) inches, find the height (altitude) of the triangle.

Solve the equation and simplify your answer. $$6 x+9=-6 x$$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Mechanics 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.