Problem 56 - Elementary and Intermediate Algebra

Question

Multiply. $$ 9(6 m+7) $$

Step-by-Step Solution

Answer

54m + 63

Step 1 - Distribute the Constant

Apply the distributive property, which states that for any numbers `a`, `b`, and `c`, the equation \( a(b + c) = ab + ac \) holds true. Here, multiply 9 by each term inside the parentheses: \( 9 \times 6m \) and \( 9 \times 7 \).

Step 2 - Multiply Each Term

Perform the multiplication: \( 9 \times 6m = 54m \) and \( 9 \times 7 = 63 \).

Step 3 - Combine the Results

Combine the results of the multiplication to get the final expression: \( 54m + 63 \).

Key Concepts

Multiplication

Multiplication is one of the basic arithmetic operations where a number is added to itself a specified number of times. For example, multiplying 9 by 6 means adding 9 six times: 9 + 9 + 9 + 9 + 9 + 9, which equals 54. In algebra, we use multiplication to simplify expressions by distributing a number across terms inside parentheses.
In our example, we have 9 multiplied by each term inside the parentheses: 6m and 7. This multiplication helps us break down expressions into more manageable pieces and prepares them for further simplification.

Algebraic Expressions

Algebraic expressions are combinations of variables, constants, and arithmetic operations. In our exercise, the algebraic expression is 6m + 7, where 6m is a term with a variable (m) and a constant (6), and 7 is just a constant.
Understanding algebraic expressions is crucial because it allows us to work with unknown values and perform operations like addition, subtraction, multiplication, and division with ease.
These expressions form the building blocks for solving equations, which are statements that two expressions are equal.

Simplifying Algebraic Expressions

Simplifying algebraic expressions means making them easier to work with by combining like terms and performing arithmetic operations where possible. In the given exercise, we use the distributive property to simplify the expression 9(6m + 7) as follows:

• First, we distribute the 9 to both terms inside the parentheses.
• Next, we perform the multiplication for each term: 9 times 6m and 9 times 7.

After carrying out these operations, we get the simplified form, 54m + 63, which is easier to handle in further mathematical processes.

Combining Like Terms

Combining like terms is a critical step in simplifying algebraic expressions. Like terms are terms that have the same variables raised to the same power. In our example, the terms 6m and 7 are different because one has the variable 'm' and the other does not, so they cannot be combined directly.

• When we use the distributive property in the exercise, we end up with terms that are already as simplified as possible: 54m and 63.
• There are no further like terms to combine since 54m and 63 are distinct terms (one with a variable and one constant).

This step finalizes the simplification process and ensures our expression is as straightforward as possible: 54m + 63.