Problem 8 - Prealgebra

Question

Use the Distributive Property to write each expression as an equivalent algebraic expression. $$8(m+4)$$

Step-by-Step Solution

Answer

The expression \(8(m + 4)\) becomes \(8m + 32\) using the Distributive Property.

Step 1: Identify the Expression

The expression given is \(8(m + 4)\). This expression includes a number outside the parenthesis, which signifies multiplication with the terms inside the parenthesis.

Step 2: Apply the Distributive Property

The Distributive Property states that \( a(b + c) = ab + ac \). Apply this property to \(8(m+4)\) by multiplying \(8\) by each term inside the parenthesis.

Step 3: Multiply the First Term

Multiply \(8\) by \(m\) which becomes \(8m\). This is the first part of the converted expression.

Step 4: Multiply the Second Term

Multiply \(8\) by \(4\) which becomes \(32\). This is the second part of the converted expression.

Step 5: Combine the Terms

Add the resulting terms from the distribution to get \(8m + 32\). This is the equivalent algebraic expression after applying the Distributive Property.

Key Concepts

Equivalent Expressions

Equivalency in algebra means that two expressions, despite possibly looking different, will always yield the same result when the variables are substituted with the same values.
For instance, in the expression given

• \(8(m + 4)\)
• when expanded using the Distributive Property becomes \(8m + 32\)

Both expressions are equivalent because they will always produce the same numerical value for any given value of \(m\).
Understanding equivalent expressions is crucial because it allows for flexibility in solving, simplifying, or comparing algebraic equations.
It also helps in identifying simpler forms of complex equations, making them easier to manage.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and arithmetic operations. The expression \(8(m + 4)\) is a classic example. It consists of:

• The number 8, which acts as a coefficient.
• The variable \(m\), which can take any real value.
• The constant 4, which stands independently from the variable.

Algebraic expressions can be as simple as a single variable or number, like \(x\) or 5, or as complex as polynomials.

They are foundational to algebra as they form the basis for equations, inequalities, and functions. A strong understanding of how to manipulate and simplify them is key to mastering algebra. Transforming algebraic expressions, such as using the Distributive Property, makes it easier to solve algebraic problems by revealing a more straightforward form.

Multiplication in Algebra

Multiplication in algebra is similar to multiplication in arithmetic, but it involves variables alongside numbers.
In the expression \(8(m + 4)\), the multiplication involves distributing the 8 across both terms within the parenthesis:

• The 8 is multiplied by the variable \(m\) to yield \(8m\).
• Simultaneously, the 8 is multiplied by the number 4 to yield 32.

These steps highlight the handling of each component in the algebraic term, maintaining the equality of expressions throughout. Understanding multiplication involving variables is crucial as it extends arithmetic principles to algebraic calculations. It helps in expanding, simplifying, and even solving equations, turning more complex expressions into manageable calculations.