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Chapter 1: Problem 57

Multiply. $$ 5(r+2+3 t) $$

Short Answer

Expert verified
5r + 10 + 15t

Step by step solution

01

Distribute the 5

Distribute the 5 to each term inside the parentheses. This means you will multiply 5 by each term individually: \[ 5(r + 2 + 3t) = 5 \times r + 5 \times 2 + 5 \times 3t \]
02

Perform the multiplication

Now, perform the multiplication for each term:\[ 5 \times r = 5r \]\[ 5 \times 2 = 10 \]\[ 5 \times 3t = 15t \]
03

Combine the results

Combine all the products from the previous step:\[ 5r + 10 + 15t \]

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Key Concepts

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

Distribution
The distribution property allows us to multiply a single term by each term within parentheses. This step is essential in algebra because it helps us handle and simplify expressions effectively.
For example, in the exercise, we have the expression \( 5(r+2+3t) \). To distribute the 5, we need to multiply it by each term inside the parentheses:
    \[ 5 \times r = 5r \] \[ 5 \times 2 = 10 \] \[ 5 \times 3t = 15t \]
Practicing distribution is crucial because it鈥檚 a fundamental algebraic skill used to simplify expressions and solve equations.
Multiplication
Multiplication is the next step after distributing when simplifying an algebraic expression. Here, you multiply constants and variables.
Let's break down what happens after distribution in our example:
    \[ 5 \times r = 5r \] This means we have 5 times the variable \( r \). Similarly, performing the other multiplications:
      \[ 5 \times 2 = 10 \] \[ 5 \times 3t = 15t \]

    It鈥檚 important to understand each multiplication separately to avoid mistakes, especially when variables and numbers are involved.
Combining Like Terms
Once all terms have been multiplied and distributed correctly, we need to combine like terms to simplify the expression further. In the given exercise \(5(r + 2 + 3t)\), the expression after distribution and multiplication is \(5r + 10 + 15t\).
Combining like terms means to add or subtract terms that have the same variable part. However, in our simplified expression \(5r + 10 + 15t\), each term is distinct:
    \(5r\) is a term with the variable \(r\)
    \(10\) is a constant term with no variable
    \(15t\) is a term with the variable \(t\)
Since there are no like terms to combine, our final simplified expression remains \(5r + 10 + 15t\).
Understanding how to combine like terms is vital because it helps reduce expressions to their simplest form and makes evaluating and solving them easier.

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