/*! 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 46 Find the GCF of each set of mono... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 46

Find the GCF of each set of monomials. $$32 m n^{2}, 16 n, 12 n^{3}$$

Short Answer

Expert verified
GCF is \(4n\).

Step by step solution

01

Identify Factors

List out the prime factors and variables for each monomial. For \(32mn^2\), the factors are \(2^5, m, n^2\).For \(16n\), the factors are \(2^4, n\).For \(12n^3\), the factors are \(2^2, 3, n^3\).
02

Find Common Factor

Identify the common factors between all the monomials. The only common prime factor among all is \(2\), and the smallest power is \(2^2\). The only common variable factor is \(n\) with the smallest power \(n^1\).
03

Calculate the GCF

Multiply the common factors together to find the GCF:\[ GCF = 2^2 imes n = 4n \].

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.

Monomials
Monomials are single mathematical expressions that consist of numbers, variables, or a combination of both, linked together by multiplication. They are characterized by having no addition or subtraction operators within them. A few examples of monomials include:
  • \( 5x \)
  • \( 3y^2z \)
  • \( 7 \)
Monomials are important in algebra because they are the building blocks for more complex expressions like polynomials. When breaking down a monomial to find the greatest common factor (GCF), we analyze both the numerical and variable components. Understanding each part of the monomial will help when simplifying or factoring larger expressions.
Prime Factorization
Prime factorization is the process of breaking down a number into a set of prime numbers that multiply together to create the original number. A prime number is defined as a number greater than 1 that has no divisors other than 1 and itself. For instance:
  • The prime factorization of \(32\) is \(2^5\).
  • The prime factorization of \(16\) is \(2^4\).
  • The prime factorization of \(12\) is \(2^2 \times 3\).
When finding the GCF of monomials, each component is broken down into its prime factors. By comparing these prime factors, we identify common elements that exist in all the numbers involved. These common elements are then used to calculate the GCF, as demonstrated with our step-by-step solution.
Variables
Variables in monomials represent unknown values and are often denoted by letters such as \( n, m, \) or \( x \). In algebraic expressions, variables can have exponents, which indicate the number of times a variable is multiplied by itself. Consider the following examples:
  • In \( n^2 \), the variable \( n \) is squared, meaning it is multiplied by itself once.
  • In \( m \), the variable \( m \) is to the power of 1, meaning it appears as it is without multiplication.
When finding the GCF, we must consider the variable components and their respective powers. The smallest power of a common variable is taken to contribute to the GCF, as seen in the example where \( n^1 \) is chosen as the common factor for the variables.
Common Factors
Common factors are numbers or variables that are present in all the terms of a set of expressions. When determining the greatest common factor of a set of monomials, identifying common factors is critical. This involves a careful look at both the numerical and variable factors of each monomial. Steps to find common factors include:
  • List the factors of each monomial.
  • Compare and determine which factors are present in all the expressions.
  • Choose the common factor with the smallest exponent in terms of both numbers and variables.
In the original solution of finding the GCF for \(32mn^2\), \(16n\), and \(12n^3\), we identified that the common numerical factor was \(2^2\) and the common variable factor was \(n^1\). Multiplying these gave us the GCF of \(4n\). Remember, the GCF is the product of all common factors.

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

Find each equivalent measure. $$1 \mathrm{m}=\underline{?} \mathrm{cm}$$

CHALLENGE Is it true that \(\frac{23}{53}=\frac{23}{53}\) or \(\frac{2}{5} ?\) Explain why or why not.

Find the GCF of each set of numbers or monomials. $$25 a, 30 b$$

Find each product. $$11.64 \times 0.001$$

Find each equivalent measure. $$1 \text { day }=\underline{?} \mathrm{h}$$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

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.