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Chapter 6: Problem 3

In the following exercises, find the greatest common factor. $$ 12 m^{2} n^{3}, 30 m^{5} n^{3} $$

Short Answer

Expert verified
6 m^2 n^3

Step by step solution

01

Find the GCF of the Coefficients

Determine the greatest common factor (GCF) of the coefficients of the given terms. The coefficients are 12 and 30. The factors of 12 are 1, 2, 3, 4, 6, and 12. The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30. The greatest common factor of these coefficients is 6.
02

Find the GCF of the Variables

Compare the variables in each term. For the variable \(m\), the lowest power is \(m^2\) since we have \(m^2\) and \(m^5\). For variable \(n\), both terms have \(n^3\), so we use \(n^3\). Combine these to find the GCF of the variables.
03

Combine the GCF of Coefficients and Variables

Multiply the GCF of the coefficients (6) by the GCF of the variables (\(m^2 n^3\)). The GCF of the terms \(12 m^2 n^3\) and \(30 m^5 n^3\) is \(6 m^2 n^3\).

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Key Concepts

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

Coefficients
Coefficients are the numerical parts of terms that include variables. In the given problem, the coefficients are the numbers 12 and 30 in the terms 12m虏n鲁 and 30m鈦祅鲁 respectively. Finding the greatest common factor (GCF) of coefficients helps simplify problems involving multiple terms.
To determine the GCF of coefficients:
  • List out the factors of each coefficient.
  • For 12: the factors are 1, 2, 3, 4, 6, and 12.
  • For 30: the factors are 1, 2, 3, 5, 6, 10, 15, and 30.
By comparing these lists, we see that the largest common factor between the two sets is 6. This step is crucial for simplifying expressions by determining shared numerical values.
Variables
Variables are symbols used to represent unknown values or to generalize numbers in equations. In the example problem, we analyze the variables 'm' and 'n'.
When finding the GCF of the variables, consider the following:
  • For variable 'm': We are given the terms m虏 and m鈦�. The lowest power (exponent) is m虏.
  • For variable 'n': Both terms have the same power, n鲁.
Thus, the GCF for the variables in the equation is m虏n鲁. This indicates the smallest degree to which each variable appears in all terms, helping us find the commonality.
Factoring
Factoring is the process of breaking down an expression into simpler terms, or factors, that when multiplied together give the original expression. It's a pivotal skill in algebra.
To find the GCF of an entire term, combine the GCF of the coefficients and the GCF of the variables. Here鈥檚 what we determined:
  • The GCF of the coefficients is 6.
  • The GCF of the variables is m虏n鲁.
Thus, the combined GCF of the expressions 12m虏n鲁 and 30m鈦祅鲁 is obtained by multiplying these factors: 6m虏n鲁.
Factoring plays a critical role in simplifying expressions and solving equations, making complex problems more manageable.

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