Problem 105 - Elementary and Intermediate Algebra

Question

Simplify each expression, if possible. $$ 9(7 m) $$

Step-by-Step Solution

Answer

The simplified expression is \(63m\).

Step 1: Understand the Expression

The expression given is \(9(7m)\). This is a multiplication problem where 9 is being multiplied by the product of 7 and \(m\).

Step 2: Apply Associative Property

According to the associative property, you can regroup numbers in multiplication. Therefore, \(9(7m)\) can be rewritten as \((9 \times 7) \times m\).

Step 3: Multiply the Constants

Calculate the product of the constants 9 and 7. \(9 \times 7 = 63\).

Step 4: Simplify the Expression

Substitute back into the expression to obtain the simplified form: \(63m\).

Key Concepts

Associative Property

The associative property is a fundamental principle that focuses on grouping numbers in operations. It is mostly applicable to addition and multiplication. This property states that the way numbers are grouped in an expression does not change the result. For instance, when dealing with the expression 9(7m), the associative property allows us to regroup it as \((9 \times 7) \times m\). The associative property helps simplify calculations by allowing us to first address the products or sums that might be easier to compute, without altering the final outcome.

Multiplication

Multiplication is one of the basic arithmetic operations and is essentially repeated addition. When multiplying numbers, we are combining groups of equal size. In algebra, multiplication often involves variables and constants. In the expression 9(7m), multiplication is used to distribute the 9 across the product 7m. Here's how it works:

• Multiply the constants first: Compute 9 times 7, which equals 63.
• Combine with variables: Attach the variable \(m\) to the product of the constants to get \(63m\).

By following these steps, you can efficiently simplify algebraic expressions.

Algebraic Expressions

Algebraic expressions are mathematical phrases that include numbers, variables, and operations. They do not have an equality sign like equations do. Consider the expression 9(7m); it is composed of constants (9 and 7) and a variable \(m\). Variables are symbols used to represent unknown values or values that can change. In algebra, it is crucial to learn how to combine and simplify these elements.

• Simplification: This involves combining like terms and using properties like the associative property to make the expression as concise as possible. In this case, simplifying 9(7m) results in 63m.
• Understanding Components: Recognizing the different parts of an expression, such as coefficients (in this case 63 is a coefficient to \(m\)), is vital for manipulation and simplification processes.

Understanding algebraic expressions helps in solving equations and real-world problems.