Problem 25 - Prealgebra

Question

Use the Distributive Property to write each expression as an equivalent expression. Then evaluate it. $$-5(8-4)$$

Step-by-Step Solution

Answer

The equivalent expression is \(-5\times8 + (-5)\times(-4)\), which evaluates to \(-20\).

Step 1: Identify the Expression

The given expression is \(-5(8-4)\). Our task is to write it using the Distributive Property and then evaluate the expression.

Step 2: Apply the Distributive Property

The Distributive Property states that \(a(b+c) = ab + ac\). Apply this to \(-5(8-4)\), interpreted as \(-5\times(8 + (-4))\). This becomes \(-5\times 8 + (-5)\times(-4)\).

Step 3: Simplify Each Term

Calculate each term individually: \(-5\times 8 = -40\) and \(-5)\times(-4) = 20\).

Step 4: Combine the Results

Add the results from the previous step: \(-40 + 20\). This simplifies to \(-20\).

Step 5: Final Evaluation

We have rewritten the expression using the Distributive Property and evaluated it, resulting in \(-20\).

Key Concepts

Equivalent Expressions

The idea of equivalent expressions is fundamental in algebra. It ensures that we can manipulate expressions without changing their value. In the exercise provided, we are asked to use the Distributive Property to rewrite the expression

• This property allows us to break down or expand expressions, thus transforming \(-5(8-4)\) into an equivalent format without altering its true value.
• When the Distributive Property is correctly applied, like in \(-5 \times 8 + (-5) \times (-4)\), it confirms that equivalent expressions result in the same evaluation at the end."

These equivalent transformations are particularly helpful when simplifying algebraic expressions for easier computation or solving equations. Recognizing that these expressions can look different yet convey the same value is key to mastering algebra.

Integer Operations

Integer operations form the basis of many calculations in algebra, and these include addition, subtraction, multiplication, and division. In our example, we use integer multiplications and additions.

• The expression \(-5 \times 8\) involves an integer operation, resulting in \(-40\).
• Similarly, the operation \(-5 \times (-4)\) yields \(+20\) due to the rules of multiplying two negative numbers.

Understanding these operations is crucial. A negative times a positive gives a negative, while a negative times a negative gives a positive result. Knowing these rules helps in predicting and verifying results while simplifying expressions and working with numerical problems.

Simplifying Expressions

Simplifying expressions is a process that aims to present expressions in their most compact and efficient form. After using the Distributive Property, the next step in our example was to simplify the resulting expression.

• Starting with breaking down the calculations \(-5 \times 8 + (-5) \times (-4)\), we calculate each to get \(-40 + 20\).
• The final step involves combining these results to obtain a single value \(-20\).

This process of reducing expressions allows mathematicians and students alike to understand and solve problems more efficiently. By simplifying expressions to their simplest form, we uncover the essential value or terms of the expression, revealing the core mathematical relationships at play.