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Chapter 10: Problem 27

Determine the greatest common factor. \(5 a b^{3}\) and \(10 a^{2} b\)

Short Answer

Expert verified
The GCF is \( 5ab \).

Step by step solution

01

Prime Factorization of Coefficients

Factorize the coefficients of each term: For 5: \[ 5 = 5 \] For 10: \[ 10 = 2 \times 5 \]
02

Factorize the Variables

Factorize the variable parts of each term: For \(5 a b^{3}\): \[ 5 a b^{3} = 5 \times a \times b \times b \times b \] For \(10 a^{2} b\): \[ 10 a^{2} b = 2 \times 5 \times a \times a \times b \]
03

Identify the Common Factors

Identify the common factors in both terms: The common factor for the coefficients: 5 The common factor for the variables (taking the lowest power of each variable): For 'a': \( a \) For 'b': \( b \)
04

Multiply the Common Factors

Multiply all the common factors to find the greatest common factor (GCF): \[ GCF = 5 \times a \times b = 5ab \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Prime Factorization
Prime factorization, a method that breaks down a number into its smallest prime factors, is essential when finding the greatest common factor (GCF).

Prime numbers are those which have no other divisors other than 1 and themselves, such as 2, 3, 5, and 7.

For example, in our exercise, we factorized the coefficients:
  • For 5: \[ 5 = 5 \]
  • For 10: \[ 10 = 2 \times 5 \]
Identifying these factors allows us to compare and find common prime factors between different numbers, which is crucial in determining the GCF.
Variables
Variables represent unknown values and can be any letter, such as 'a' or 'b'.
In expressions, variables can have exponents that indicate how many times the variable is multiplied by itself.

In our example, each term was factorized:
For \[5ab^3 \]: \[ 5 \times a \times b \times b \times b \]
For \[10a^2b \]: \[ 2 \times 5 \times a \times a \times b \]

Breaking down each variable and its exponents lets us easily compare and find the common variable factors with their respective lowest powers.
Coefficients
Coefficients are the numerical part of the terms.
They multiply the variables.

In our example, the coefficients are 5 and 10.
When determining the GCF, it's essential to factorize the coefficients as well:
  • 5 can be factorized as \[ 5 = 5 \]
  • 10 can be factorized as \[ 10 = 2 \times 5 \]
In this way, we identify the common factor of 5 between these coefficients efficiently.
Common Factors
Common factors are numbers or variables that appear in each term.
To determine the GCF, we look for the lowest power of each common variable and multiply them together with the common factor of the coefficients.

In our case, the common factors are:
  • For the coefficients: 5
  • For the variables: 'a' (smallest power is \(a^1\)) and 'b' (smallest power is \(b^1\))
By combining these, the GCF is determined as \[ 5 \times a \times b = 5ab \].
This GCF represents the greatest value that each of the original expressions can be evenly divided by.

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