Problem 115 Create factor trees for each num... [FREE SOLUTION]

91影视

Prealgebra

David Arnold

$Math Studyset 91影视 Explanations$ Math

2020 Edition

Chapter 1: Problem 115

Create factor trees for each number. Write the prime factorization for each number in compact form, using exponents. $$12$$

Short Answer

Expert verified

The prime factorization of 12 is 2^2 x 3.

Step by step solution

Start the Factor Tree

Begin by identifying a pair of factors for 12. You can choose any pair of numbers that multiply to give 12. Let's start with 2 and 6.

Factor the Next Level

Consider the number 6 from the previous step. Find two numbers that multiply to give 6. We can use 2 and 3.

Complete the Factor Tree

Now we have broken down 12 into prime factors: 2, 2, and 3. The factor tree for 12 is complete when all the branches end in prime numbers: 12 -> 2 x 6 -> 2 x (2 x 3).

Write the Prime Factorization

The prime factors of 12, as determined from the factor tree, are 2, 2, and 3. Write these factors in compact form using exponents: 12 = 2^2 x 3.

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Key Concepts

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

Factor Trees

Factor trees are a systematic method to break down a whole number into its prime factors. Think of a factor tree as a way to "peel off" layers of a number until you鈥檙e only left with its building blocks, or prime numbers. Here's a simple breakdown of how to use factor trees:

Begin with the number you want to factorize, in this case, 12.
Select two factors of the number. These are any two numbers that multiply to give the original number. For 12, a common choice is 2 and 6, but you could also start with 3 and 4.
Continue breaking down any composite (non-prime) numbers. For instance, the number 6 can be further broken down into the factors 2 and 3.
Repeat this process until all branches of the tree end in prime numbers. For instance, in our example, the number 12 becomes 2 x 2 x 3.

Factor trees provide a visual aid that makes it easier to see the breakdown of numbers into their prime components. By using factor trees, complex calculations can become more straightforward.

Multiplication

In the context of factor trees and prime factorization, multiplication plays a crucial role. It's the operation that connects the factors back together to form the original number. Here's how you can think about it in the process of factorization:

Multiplying two numbers gives you a product, which can be further explored to find more factors if needed.
As we multiply the prime factors derived from a factor tree, such as 2 x 2 x 3 for the number 12, we see how these prime numbers multiply back to the original number: 2 x 2 = 4, and 4 x 3 = 12.
This repeated use of multiplication ensures the accuracy of the factorization process.

By practicing multiplication within the context of factor trees, you not only learn to 'reverse' the process of factorization but also gain confidence in understanding how numbers relate to one another.

Exponents

Exponent notation is a way of writing repeated multiplication compactly. It streamlines the presentation of number factors by showing how many times a prime number is used in the factorization. Here's how it works:

An exponent tel艂s you how many times a number, called the base, is multiplied by itself. In the factorization of 12, the base is 2, and it appears twice.
You write this repeated multiplication compactly as 2^2. This means 2 is used as a factor twice in the multiplication process.
Using exponents simplifies expressions, making complex calculations more manageable.

When we look at how 12 is factorized into 2 x 2 x 3 and write it as 2^2 x 3, we're seeing the power of using exponents. They not only make the expression shorter but also indicate the multiplicity of each factor, providing a clearer view of the number鈥檚 structure.

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91影视

Create factor trees for each number. Write the prime factorization for each number in compact form, using exponents. $$12$$

Short Answer

Step by step solution

Start the Factor Tree

Factor the Next Level

Complete the Factor Tree

Write the Prime Factorization

Key Concepts

Factor Trees

Multiplication

Exponents

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