Problem 16 Find the greatest common factor ... [FREE SOLUTION]

Chapter 5: Problem 16

Find the greatest common factor for each list of terms. \(a^{4} b^{5}, a^{3} b\)

Short Answer

Expert verified

The GCF is \(a^3 b\).

Step by step solution

- Identify common factors

First, identify the common factors in each term. The terms are: \(a^4 b^5\) and \(a^3 b\). Both terms have the variables \(a\) and \(b\).

- Find the lowest power of each common variable

Next, find the lowest power of each common variable. For the variable \(a\), the powers are \(4\) and \(3\). The lowest power is \(3\). For the variable \(b\), the powers are \(5\) and \(1\). The lowest power is \(1\).

- Combine the lowest powers

Combine these lowest powers to form the greatest common factor (GCF). Thus, the GCF is \(a^3 b^1\).

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.

common factors

When we talk about common factors in algebra, we鈥檙e referring to the variables or numbers that the terms share. Think of it like finding common ingredients in different recipes. If two terms have the same variable, like both having an 'a' or a 'b', those variables are their common factors. Remember, we can only combine or use these variables if both terms have them. If one term has a variable that the other doesn't, that variable isn't a common factor.

lowest power

The concept of the lowest power is crucial when finding the greatest common factor (GCF). Each variable in a term can have what we call an exponent or power. This power tells us how many times the variable is multiplied by itself. For instance, in the term \(a^4 \), the '4' is the power of the variable 'a'. When determining the greatest common factor, we look for the smallest power of each variable that appears in all the terms. For example, between \(a^4 \) and \(a^3 \), the lowest power of 'a' is '3'. We use this smaller power because it's the most that both original terms can share equally.

variables

Variables are symbols that represent numbers in algebra. They can stand for almost anything but are most often used to generalize mathematical rules and operations. For example, in the expression \(a^4 b^5\), 'a' and 'b' are variables. Each variable can have different powers (or exponents), which indicate how many times the variable multiplies itself. When we find the greatest common factor, we look for each variable they have in common. Then we take the lowest power of each to combine them. This helps us simplify expressions and solve equations more easily. For the original terms given, the GCF is therefore \(a^3 b^1\), using the lowest powers of each variable they both share.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Find the greatest common factor for each list of terms. \(a^{4} b^{5}, a^{3} b\)

Short Answer

Step by step solution

- Identify common factors

- Find the lowest power of each common variable

- Combine the lowest powers

Key Concepts

common factors

lowest power

variables

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Statistics

Pure Maths

Decision Maths

Probability and Statistics

Geometry

Study anywhere. Anytime. Across all devices.