Problem 53 - Elementary and Intermediate Algebra

Question

Multiply. $$ 8(3 x+5 y) $$

Step-by-Step Solution

Answer

24x + 40y

Step 1 - Distribute the Multiplication

Apply the distributive property of multiplication to expand the expression. You need to multiply each term inside the parentheses by the factor outside. In this case, multiply 8 by both terms inside the parentheses, which are 3x and 5y.

Step 2 - Multiply Each Term

First, multiply 8 by 3x: \[ 8 \times 3x = 24x \]Second, multiply 8 by 5y: \[ 8 \times 5y = 40y \]

Step 3 - Combine the Products

Combine the results from Step 2 to form the expanded expression. The expanded form will be the sum of the products:\[ 24x + 40y \]

Key Concepts

multiplication

Multiplication is one of the four basic mathematical operations. It involves finding the total number of objects in a given number of equal-sized groups. In the context of our exercise, we use multiplication to scale up the values of algebraic terms. For example, to multiply 8 by 3x:

\[ 8 \times 3x = 24x \]

This means 3x is added together 8 times, resulting in 24x. Similarly, 8 multiplied by 5y becomes 40y.

algebraic expressions

Algebraic expressions are mathematical phrases that include numbers, variables, and operations. In the given exercise, we have the expression \[ 8(3x + 5y) \]

Here, 3x and 5y are algebraic terms. The expression inside the parentheses is called a binomial because it contains two terms. To work with algebraic expressions, you follow the same arithmetic operations as you would with regular numbers, including addition, subtraction, multiplication, and division.

expanding expressions

Expanding expressions means simplifying an expression by removing parentheses. You do this by using the distributive property of multiplication over addition or subtraction. In our exercise, to expand \[ 8(3x + 5y) \]

You distribute 8 to both 3x and 5y:

• First, multiply 8 by 3x to get 24x.
• Then, multiply 8 by 5y to get 40y.

Finally, you combine the terms to get the expanded form:

\[ 24x + 40y \]

mathematical operations

Mathematical operations include addition, subtraction, multiplication, and division. These operations follow specific rules known as the order of operations or PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).

In our exercise, we primarily use multiplication to expand the expression. Understanding how to apply these operations correctly is crucial for solving algebraic expressions and other math problems effectively.