Problem 39 - Algebra 1: Concepts and Skills

Question

Simplify the expression. \((-4 \cdot 6)^{2}\)

Step-by-Step Solution

Answer

The simplified form of the expression \((-4 \cdot 6)^{2}\) is 576.

Step 1: Perform the Multiplication

In the expression \((-4 \cdot 6)^{2}\), the first step is to perform the multiplication inside the brackets, i.e., multiply -4 by 6 to get -24. So, the expression becomes \((-24)^{2}\).

Step 2: Perform the Squaring Operation

Next, square the result that was obtained in the previous step, i.e., square -24. The square of -24 equals to 576.

Key Concepts

Understanding Multiplication in Expressions

Multiplication is one of the fundamental operations in mathematics. It involves finding the product of two numbers. In this expression, \(-4 \cdot 6\), we're multiplying -4 by 6.

This can be visualized as adding -4 together six times:

• -4 + -4 + -4 + -4 + -4 + -4
• The result is -24.

The negative sign here simply means you're moving in the opposite direction on a number line or having a deficit.

Multiplying a negative number by a positive number always results in a negative product.

Exploring Exponents

Exponents represent repeated multiplication. When you have a number raised to an exponent, you multiply that number by itself as many times as the exponent tells you.

For example, in \((-24)^{2}\), you're squaring -24. This means:

• \((-24) \times (-24)\)

Squaring a negative number results in a positive product because a negative times a negative equals a positive.
Therefore, \((-24)^2 = 576\).
Understanding exponents involves knowing how to break down the multiplication process step by step, so you don't miss out on nuances like sign changes.

Mastering Order of Operations

The order of operations is crucial in simplifying expressions correctly. It dictates the sequence in which operations should be performed to arrive at the correct answer.

The standard order is:

• Parentheses (or brackets)
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

In this problem, \((-4 \cdot 6)^{2}\) follows the order of operations:

• Calculate inside the parentheses first: \(-4 \cdot 6 = -24\).
• Next, handle the exponent to square the result: \(-24\) becomes \(576\).

By following this order, you avoid common mistakes and ensure accurate results every time.