/*! 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 39 Simplify each fraction. $$ \... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 39

Simplify each fraction. $$ \frac{1+x^{-1}-12 x^{-2}}{1-x^{-1}-20 x^{-2}} $$

Short Answer

Expert verified
\( \frac{x - 3}{x - 5} \)

Step by step solution

01

- Factorize the numerator

First, factorize the numerator \(1 + x^{-1} - 12x^{-2}\). Rewrite it as \( (x^{-2})(x^2 + x - 12)\). Find factors of \( x^2 + x - 12 \), which can be factored to \( (x + 4)(x - 3) \). Thus, the numerator becomes \( x^{-2}(x + 4)(x - 3) \).
02

- Factorize the denominator

Now, factorize the denominator \(1 - x^{-1} - 20x^{-2}\). Rewrite it as \( (x^{-2})(x^2 - x - 20)\). Find factors of \( x^2 - x - 20 \), which can be factored to \( (x - 5)(x + 4) \). Thus, the denominator becomes \( x^{-2}(x - 5)(x + 4) \).
03

- Simplify the fraction

Now, simplify the fraction by canceling common factors in the numerator and the denominator. The common factors are \( x^{-2} \) and \( x + 4 \). Canceling these factors results in: \[ \frac{x^{-2}(x + 4)(x - 3)}{x^{-2}(x - 5)(x + 4)} = \frac{x - 3}{x - 5} \]

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.

Factoring Expressions
Factoring expressions is a fundamental skill in algebra. It helps in simplifying complex algebraic expressions or equations. Factoring involves breaking down a complex expression into simpler factors that, when multiplied, give the original expression.

For example, we factor the quadratic expression \(x^2 + x - 12\) into \((x + 4)(x - 3)\). This is because the product of \4 \times (-3) = -12\, and the sum of \4 + (-3) = 1\.

To factorize an expression:
  • Look for common factors in the terms.
  • Use formulas like the difference of squares or sum/product of roots for quadratic expressions.
  • Recheck by expanding your factors to ensure correctness.
Rational Expressions
Rational expressions are fractions where the numerator and the denominator are polynomials. Simplifying rational expressions means to reduce them to the simplest form by canceling common factors.

Consider the fraction \(\frac{1 + x^{-1} - 12x^{-2}}{1 - x^{-1} - 20x^{-2}}\). Here, expressions in both the numerator and the denominator can be factored to help simplify the fraction.

A rational expression is simplified when:
  • The numerator and the denominator are factored completely.
  • Common factors in the numerator and the denominator are canceled out.
  • It is impossible to further reduce the expression.

In our example, after factoring we canceled \(x^{-2}\) and \(x + 4\) from both numerator and denominator, resulting in \frac{x - 3}{x - 5}\.
Algebraic Manipulation
Algebraic manipulation involves rearranging equations and expressions using algebraic rules. This becomes crucial when simplifying expressions or solving equations.

Steps involved for our exercise include:
  • Identifying and rewriting negative exponents: For instance, converting \(x^{-1}\ to\ \frac{1}{x}\)
  • Factoring both the numerator and denominator: Recognizing patterns like quadratics that can be factored.
  • Canceling out common terms: Simplify the fractions by removing identical factors in both the numerator and denominator.

Let's apply these:

  1. \(x^{-2}(x^2 + x - 12)\) in the numerator becomes \(x^{-2}(x + 4)(x - 3)\).
  2. \(x^{-2}(x^2 - x - 20)\) in the denominator becomes \(x^{-2}(x - 5)(x + 4)\).
  3. Now, cancel the common factors to get \(\frac{x - 3}{x - 5}\).

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

Multiply or divide as indicated. Write each answer in lowest terms. See Section 1.1 $$ \frac{7}{12} \div \frac{15}{4} $$

Multiply or divide. Write each answer in lowest terms. See Examples \(3,6,\) and 7 . $$\frac{2 m^{2}-5 m-12}{m^{2}-10 m+24} \cdot \frac{m^{2}-9 m+18}{4 m^{2}-9}$$

Solve each formula for \(k\) $$ y=\frac{k}{x^{2}} $$

Multiply or divide as indicated. Write each answer in lowest terms. See Section 1.1 $$ \frac{10}{3} \div \frac{5}{6} $$

Solve each equation for \(k\) $$ 200=15 k $$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Calculus

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.