/*! 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 11 For Problems \(9-50\), simplify ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 11

For Problems \(9-50\), simplify each rational expression. $$ \frac{18 a^{2}}{45 a b} $$

Short Answer

Expert verified
\(\frac{2a}{5b}\)

Step by step solution

01

Factor the Numerator and Denominator

We start by identifying the common factors in the numerator and denominator. The numerator is \(18a^2\) and the denominator is \(45ab\). Factoring each, we get:- \(18a^2 = 2 \times 3^2 \times a^2\)- \(45ab = 3^2 \times 5 \times a \times b\)
02

Identify and Cancel Common Factors

Both the numerator and the denominator share the common factors \(3^2 = 9\) and \(a\). Therefore, we can cancel out these common factors from both:\[ \frac{2 \times 3^2 \times a^2}{3^2 \times 5 \times a \times b} = \frac{2 \times a}{5 \times b} \]
03

Simplify the Expression

After cancelling the common factors, no further simplification is possible. Thus, the expression is simplified to:\[ \frac{2a}{5b} \]

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 Algebraic Expressions
Factoring algebraic expressions is like simplifying a puzzle. The objective is to express a number or expression as a product of its factors.
Factors are numbers or expressions you can multiply to get the original number or expression. To factor the number 18, we look for numbers that multiply to give 18. Since 18 equals 2 times 9, and 9 is 3 times 3, we can write:
  • 18 = 2 x 3 x 3 = 2 x 3^2
For an expression like 18a虏, factor it by breaking down both the number and the expression:
  • 18a虏 = 2 x 3^2 x a x a = 2 x 3^2 x a^2
Similarly, the expression 45ab factors to:
  • 45ab = 3^2 x 5 x a x b
Remember, factoring is a critical step as it allows us to easily identify common factors between expressions.
Canceling Common Factors
Once factors are identified, the next step is to cancel common factors in both the numerator and the denominator. This process is essential in simplifying rational expressions.For instance, after factoring,
  • the numerator is: 2 x 3^2 x a^2
  • the denominator is: 3^2 x 5 x a x b
Here, both expressions have common factors of 3^2 and a. Cancel these out as they appear in both the numerator and the denominator:\[\frac{2 \times 3^2 \times a^2}{3^2 \times 5 \times a \times b} = \frac{2 \times a}{5 \times b}\]The cancellation leaves us with a simplified expression of \( \frac{2a}{5b} \).
By canceling common factors, you ensure the expression is in its simplest form. This makes further problem-solving steps easier.
Intermediate Algebra
Intermediate Algebra is the bridge between basic arithmetic and advanced algebra. It provides the tools to simplify complex expressions, work with polynomials, and solve equations.Understanding Intermediate Algebra involves:
  • Breaking down expressions into components
  • Recognizing patterns and common factors
  • Applying rules of arithmetic and algebra to simplify expressions
In our example, Intermediate Algebra techniques were used to identify common factors and reduce the expression \( \frac{18a^2}{45ab} \) to \( \frac{2a}{5b} \). This step-by-step simplification is crucial:
It helps in tackling more advanced algebraic problems by ensuring all expressions are in their simplest form. Mastering these concepts will ease your journey into higher math levels, making calculations more manageable and accurate.

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

For Problems \(1-44\), solve each equation. $$ \frac{x}{x+1}+3=\frac{4}{x+1} $$

For Problems \(1-30\), solve each equation. $$ \frac{x}{x-4}-\frac{2}{x+8}=\frac{63}{x^{2}+4 x-32} $$

For Problems \(1-30\), solve each equation. $$ \frac{-2}{3 x+2}+\frac{x-1}{9 x^{2}-4}=\frac{3}{12 x-8} $$

To travel 60 miles, it takes Sue, riding a moped, 2 hours less time than it takes Doreen to travel 50 miles riding a bicycle. Sue travels 10 miles per hour faster than Doreen. Find the times and rates of both girls.

Plane A can travel 1400 miles in 1 hour less time than it takes plane B to travel 2000 miles. The rate of plane B is 50 miles per hour greater than the rate of plane A. Find the times and rates of both planes.
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

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.