/*! 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 63 Simplify each expression, using ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 63

Simplify each expression, using only positive exponents in the answer. $$ \frac{x^{-1}+2 y^{-1}}{2 y+4 x} $$

Short Answer

Expert verified
\( \frac{1}{2 y x} \)

Step by step solution

01

- Understand the problem

Given the expression: \( \frac{x^{-1} + 2 y^{-1}}{2 y + 4 x} \), simplify it so that all exponents are positive.
02

- Rewrite negative exponents

Rewrite the terms with negative exponents using positive exponents: \( x^{-1} = \frac{1}{x} \) and \( y^{-1} = \frac{1}{y} \). So the expression becomes: \( \frac{\frac{1}{x} + 2 \frac{1}{y}}{2 y + 4 x} \).
03

- Find common denominator for the numerator

Combine the fractions in the numerator by finding a common denominator: \( \frac{1}{x} + 2 \frac{1}{y} = \frac{y}{xy} + \frac{2x}{xy} = \frac{y + 2x}{xy} \). So the expression now is: \( \frac{\frac{y + 2x}{xy}}{2 y + 4 x} \).
04

- Simplify the complex fraction

Simplify the complex fraction by multiplying the numerator by the reciprocal of the denominator: \( \frac{y + 2x}{xy} \times \frac{1}{2 y + 4 x} \). There is nothing to cancel out, so the simplified form is: \( \frac{y + 2x}{xy(2 y + 4 x)} \).
05

- Simplification

Observe that both the numerator and the denominator have a common factor of \( y + 2x \). Divide both the numerator and the denominator by this common factor to get: \( \frac{1}{2 y x} \).

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 Exponents
In algebra, negative exponents can seem tricky at first, but they follow a straightforward rule: any base raised to a negative exponent is the same as the reciprocal of the base with a positive exponent.
For example, \(x^{-1}\) is the same as \(\frac{1}{x} \), and \( y^{-1} \) is equal to \( \frac{1}{y} \).
This rule helps simplify expressions by rewriting them without negative exponents:
  • \(x^{-2} = \frac{1}{x^2} \)
  • \(a^{-3} = \frac{1}{a^3} \)
  • \(b^{-4} = \frac{1}{b^4} \)
Common Denominators
Finding a common denominator is essential for adding and subtracting fractions.
It helps combine fractions into a single fraction. Let's break down this concept using an example:
  • Consider the fractions \( \frac{1}{x} \) and \( \frac{2}{y} \).
  • To add \( \frac{1}{x} \) and \( \frac{2}{y} \), find a common denominator, which would be \( xy \).
  • Rewrite each fraction with this common denominator: \( \frac{1}{x} = \frac{y}{xy} \) and \( \frac{2}{y} = \frac{2x}{xy} \).
  • Combine these: \( \frac{y}{xy} + \frac{2x}{xy} = \frac{y + 2x}{xy} \).
Complex Fractions
A complex fraction is a fraction where the numerator, the denominator, or both, contain fractions themselves.
Simplifying a complex fraction involves a few steps:
  • Simplify the numerator and the denominator separately.
  • Combine them into a single fraction if needed by finding common denominators.
  • Multiply the numerator by the reciprocal of the denominator.
For our exercise:
  • Start with \( \frac{\frac{y + 2x}{xy}}{2y + 4x} \).
  • Rewrite by multiplying the numerator by the reciprocal of the denominator: \( \frac{y + 2x}{xy} \times \frac{1}{2y + 4x} \).
  • This gives the simplified expression: \( \frac{y + 2x}{xy(2y + 4x)} \).
After recognizing that \( y + 2x \) is a common factor, divide both the numerator and denominator by \( y + 2x \):
\( \frac{1}{2yx} \).

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 each formula or equation for the specified variable. $$ d=\frac{2 S}{n(a+L)} \text { for } S $$

Solve each formula or equation for the specified variable. $$ m=\frac{y-b}{x} \text { for } y $$

Solve each formula or equation for the specified variable. $$ \frac{3}{k}=\frac{1}{p}+\frac{1}{q} \text { for } q $$

Simplify each complex fraction. Use either method. $$ \frac{\frac{6}{x+3}+3}{\frac{9}{x+3}-3} $$

Find the slope of the line that passes through each pair of points. This will involve simplifying complex fractions. $$ \left(\frac{3}{4}, \frac{1}{3}\right) \text { and }\left(\frac{5}{4}, \frac{10}{3}\right) $$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Decision 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.