Problem 41 - Prealgebra

Question

The problems below review the material on exponents we have covered previously. Expand and simplify. $$(-3)^{2}$$

Step-by-Step Solution

Answer

The expanded and simplified result of \((-3)^2\) is 9.

Step 1: Understand the Expression

The expression \((-3)^2\) consists of a base \(-3\) and an exponent 2. The exponent indicates how many times we need to multiply the base by itself.

Step 2: Expand the Base

Since the exponent is 2, we will multiply \(-3\) by itself: \[(-3) imes (-3)\]

Step 3: Multiply the Terms

Now, multiply \(-3\) by \(-3\): \[(-3) imes (-3) = 9\]Because a negative times a negative results in a positive, the result is 9.

Step 4: Finalize the Answer

The expanded and simplified result of \((-3)^2\) is 9.

Key Concepts

Multiplication of Integers

Multiplication of integers can seem tricky when dealing with negative numbers, but once you understand the rules, it’s quite straightforward. Integers include all whole numbers and their negative counterparts. When multiplying two integers, there are specific signs rules to follow.

• If you multiply two positive integers, the result is positive.
• If you multiply one positive integer and one negative integer, the result is negative.
• If you multiply two negative integers, the result turns positive.

In the exercise \((-3)^2\), we're multiplying \(-3\) by itself, which involves two negative numbers. As per the rules, the product is positive, resulting in the number 9. It’s crucial to always remember these sign rules, as they are fundamental to working with integers in multiplication.

Base and Exponent

In math, understanding what a base and an exponent represent is key to mastering exponents. The base is the number you're repeatedly multiplying, while the exponent tells you how many times to multiply the base by itself.
For example, in \((-3)^2\), \(-3\) is the base and 2 is the exponent. This means we multiply \(-3\) by itself 2 times: \((-3) \times (-3)\). Starting with the correct setup of base and exponent is vital. It’s like setting the stage for what happens next in solving the problem.
Visualizing exponents can become intuitive once you practice separating base from exponent. Consider the exponent as a shortcut, showing repeated multiplication without having to write out every step.

Negative Numbers

Negative numbers, appearing with a minus sign before them, add an additional layer to understanding math problems. They represent values less than zero and follow particular rules when interacting with other numbers.
In multiplication, two negative numbers combine to form a positive number. This may initially seem counterintuitive, but think of it as a reversal similar to turning a negative direction back to positive. With \((-3)^2\), we started with a negative base because there are two negatives being multiplied together, the negatives cancel out, which results in a positive answer: 9.
When approaching problems with negative numbers, remember to keep a close watch on the signs. This ensures accuracy in problems involving everything from simple arithmetic to complex equations.