Problem 22 - Prealgebra

Question

Use the Distributive Property to write each expression as an equivalent expression. Then evaluate it. $$6(8-5)$$

Step-by-Step Solution

Answer

The equivalent expression is \(48 - 30\) and it evaluates to 18.

Step 1: Apply the Distributive Property

To rewrite the expression using the distributive property, multiply the number outside the parenthesis by each term inside the parenthesis. The expression is: \(6(8-5)\). Applying the distributive property gives us:\[6 \times 8 - 6 \times 5\].

Step 2: Perform the Multiplication

Now, we multiply each pair of numbers. First calculate \(6 \times 8\) and \(6 \times 5\). The calculations are: \(6 \times 8 = 48\) and \(6 \times 5 = 30\). So the expression becomes \(48 - 30\).

Step 3: Subtract to Simplify

Now subtract the two values obtained from the multiplications. Calculate \(48 - 30\) to simplify the expression into a single value. The result is \(18\).

Key Concepts

Equivalent Expressions

In mathematics, expressions that have the same value for all values of the variable(s) in them are known as equivalent expressions. Let's consider the original expression for clarity: \(6(8-5)\). By using the distributive property, we can transform this expression into another form. We multiply the 6 by each term inside the parentheses: 6 times 8 and 6 times 5. Thus, the expression is rewritten as \(6 \times 8 - 6 \times 5\).
This new form of the expression is considered equivalent to the original because, when calculated, they both yield the same numerical result. This is crucial in algebra, as it allows us to manipulate expressions to make them easier to work with or solve. Recognizing and creating equivalent expressions is a foundational skill that makes complex problems more approachable.

Multiplication

Multiplication is an arithmetic operation that combines groups of equal sizes. In the context of the distributive property, multiplication is used to distribute a term over a sum or difference inside the parentheses.
In our example, after applying the distributive property to \(6(8-5)\), we see the two multiplication tasks needed: \(6 \times 8\) and \(6 \times 5\). Firstly, perform \(6 \times 8\) to get 48, and then \(6 \times 5\) to get 30. These operations break down the expression step-by-step, making it manageable.
Understanding multiplication in distribution helps us simplify expressions and solve equations efficiently. It highlights multiplication as a repetitive addition, where \(6 \times 8\) is the same as adding 6 to itself 8 times.

Simplification

Simplification in math means reducing an expression to its most compact form without changing its value. After applying the distributive property and performing multiplication, we reach the final step of simplification.
For the expression \(6(8-5)\), after using the distributive property and calculating, we end up with \(48 - 30\). The next step is to simplify the expression by performing the subtraction.
Calculate \(48 - 30\), which yields 18. This result is a simplified expression; it cannot be reduced any further. Simplification helps in making expressions easier to work with and understand, providing a clear and concise representation of the problem at hand.