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Chapter 5: Problem 77

Multiply. $$ (a+6) 2 $$

Short Answer

Expert verified
The product is \(2a + 12\).

Step by step solution

01

Identify the Components

We are given an expression \((a+6) \times 2\).The two components here are \(a+6\)and the number \(2\). Our task is to multiply these together.
02

Distribute the Constant

The expression \((a+6) \times 2\)requires us to distribute the \(2\)across both terms inside the parentheses.This involves two operations:1. Multiply \(a\)by \(2\):\[(a \times 2) = 2a\]2. Multiply \(6\)by \(2\):\[(6 \times 2) = 12\]
03

Combine the Results

After distribution, combine the results of the two operations from Step 2.Write the expression as:\[2a + 12\] This is the simplified result of multiplying \((a+6)\) by \(2\).

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Key Concepts

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

Distributive Property
The distributive property is a fundamental principle in algebra that helps us simplify multiplication involving expressions with parentheses. It allows us to distribute, or pass out, a term across the terms inside the parentheses. In our problem, we are multiplying \((a+6) \times 2\). To apply the distributive property, you take the number outside the parenthesis (in this case, 2) and multiply it by each term inside the parentheses separately.

  • Distribute the 2 to each term in \(a+6\).
  • Multiply 2 times \(a\), resulting in \(2a\).
  • Multiply 2 times 6, giving \(12\).
Applying the distributive property ensures that every term within the parentheses is accounted for, making it easier to manipulate and simplify expressions.
Simplifying Expressions
Simplifying expressions involves reducing them to their most concise form without altering their meaning. After applying the distributive property, the next step is to combine all your results into one expression. In our example,the distributed expression is \(2a + 12\). There are no like terms to combine further, so this is the simplest form.

  • Ensure all distribution is complete.
  • Combine any like terms, although in this example, there are none.
  • Write the expression in its simplest form, \(2a + 12\).
Understanding how to simplify expressions helps in streamlining your calculations and makes problem solving more efficient and error-free.
Algebraic Expressions
Algebraic expressions are combinations of variables, numbers, and operations that represent different values. They can include:
  • variables (like \(a\)),
  • constants (like 6),
  • and operations (like addition and multiplication).
Our original expression is \((a+6) \times 2\), which involves these components.

Understanding how to manipulate algebraic expressions using rules like the distributive property is essential in algebra. It allows you to solve equations, model real-world scenarios, and perform calculations accurately. In our case, you rewrote the expression in a simplified form \(2a + 12\), making it clearer for further use or solving!

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