Problem 72 - Elementary Algebra

Question

For the following problems, use the distributive property to expand the expressions. $$ (8 m+5 n) 6 p $$

Step-by-Step Solution

Answer

Answer: The expanded form of the expression is \(48mp + 30np\).

Step 1: Identify the distributive property components

In the given expression \((8m + 5n)6p\), we can identify the components a, b, and c as follows: - a: \(6p\) - b: \(8m\) - c: \(5n\)

Step 2: Apply the distributive property

Now, we will apply the distributive property \(a(b + c) = ab + ac\) using the components we identified in the previous step. $$ (8m + 5n)6p = 6p(8m) + 6p(5n) $$

Step 3: Multiply the terms

Next, we will multiply the terms as shown: $$ 6p(8m) + 6p(5n) = 48mp + 30np $$

Step 4: Write the final expression

After applying the distributive property and simplifying the expression, we get the final expanded expression: $$ (8m + 5n)6p = 48mp + 30np $$

Key Concepts

Algebraic Expressions

An algebraic expression is a mathematical phrase that contains numbers, variables (like m or n), and operation signs (such as +, −, ×, ÷). In our exercise example, (8m + 5n)6p is an algebraic expression involving the variables m, n, and p. These expressions can represent quantities that can change or vary, which is why they're so fundamental in algebra.

Understanding how to work with algebraic expressions is essential because they are the building blocks for more complex mathematical concepts like equations and functions. The expression from the exercise includes addition inside the parentheses and multiplication outside, which encourages the use of the distributive property to expand the expression.

Simplifying Expressions

Simplifying an expression means to rewrite it in a simpler or more concise form without changing its value. One of the primary techniques for simplifying expressions is to use the distributive property. This property lets you multiply a single term by each term inside a set of parentheses.

For example, applying the distributive property to expand (8m + 5n)6p means multiplying 6p by both 8m and 5n. It transforms the expression into 48mp + 30np, which is considered simpler because it spells out the multiplication explicitly. This form is more straightforward and often makes it easier to continue with other operations such as addition or subtraction.

Elementary Algebra

Elementary algebra introduces the fundamental concepts of algebra, focusing on operations and manipulation of algebraic expressions and equations. The distributive property is a cornerstone of elementary algebra. It states that for any three numbers, variables, or terms, the equation a(b + c) = ab + ac holds true.

In the context of the given problem, elementary algebra skills like identifying terms and applying the distributive property are essential. The solution involves both recognizing the structure of the algebraic expression and knowing how to properly apply the distributive property to expand it. Ultimately, practicing these skills can lead to mastering more advanced algebraic techniques.