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Chapter 1: Problem 14

Use the distributive property to rewrite each expression. See Example 2 $$ 12(12+3) $$

Short Answer

Expert verified
Distributed expression is 180.

Step by step solution

01

Identify the Expression

The given expression is \(12(12 + 3)\). This indicates that you need to distribute the 12 across both terms inside the parentheses.
02

Apply the Distributive Property

Using the distributive property, distribute 12 to each term inside the parentheses: \(12 \times 12 + 12 \times 3\).
03

Perform the Multiplication

Calculate the products individually: \(12 \times 12 = 144\) and \(12 \times 3 = 36\).
04

Combine the Products

Add the results of the two multiplications: \(144 + 36\).
05

Final Calculation

Add the two numbers to get the final expression: \(180\). Therefore, \(12(12+3)\) simplifies to 180.

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Key Concepts

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

Multiplication
Multiplication is one of the basic arithmetic operations. In our example, you have to multiply numbers as part of simplifying the expression. Multiplication combines groups of a number to find totals. When applying multiplication in a problem like this:
  • The idea is to multiply each number you are distributing outside the parentheses by each term inside.
  • Think of it as finding the sum of repeated additions.
For example, with our problem 12 multiplied by both 12 and 3, you calculate:
  • \(12 \times 12 = 144\)
  • \(12 \times 3 = 36\).
Perform each multiplication separately before combining your answers as given by the distributive property principle.
Arithmetic Operations
Arithmetic operations include addition, subtraction, multiplication, and division. These fundamental operations allow us to perform calculations. Understanding these can help you solve practically any math problem. In our problem, we used multiplication and addition.
  • First, multiplication was used with \(12 \times 12\) and \(12 \times 3\).
  • Then, we used addition to find the total of these products.
An arithmetic sequence of operations, as seen here, converts a potentially complex problem into simpler steps that are easier to solve.
Simplifying Expressions
Simplifying expressions involves reducing an expression to its simplest form. Using the distributive property simplifies the expression \(12(12 + 3)\) by eliminating the parentheses through multiplication, and then simplifying further through arithmetic operations.
  • Start by distributing: \(12(12 + 3) = 12 \times 12 + 12 \times 3\).
  • Calculate each multiplication individually: \(12 \times 12 = 144\) and \(12 \times 3 = 36\).
  • Add the products: \(144 + 36 = 180\).
After completing these steps, the expression \(12(12 + 3)\) is simplified to the value 180. Simplifying down to a single numerical value is the goal when using the distributive property in this way.

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