Problem 8 - Algebra and Trigonometry

Question

Evaluate each expression. $$ (-3)^{2} $$

Step-by-Step Solution

Answer

The expression \((-3)^2\) evaluates to 9.

Step 1: Understand the Expression

We are given the expression \((-3)^2\). This notation means that we need to multiply -3 by itself two times.

Step 2: Square the Number

To calculate \((-3)^2\), multiply -3 by itself. This gives you: \[ (-3) \times (-3) = 9 \]

Step 3: Final Step: Combine Your Results

After calculating \((-3) \times (-3)\), we find that \((-3)^2 = 9\).

Key Concepts

Understanding Exponents

Exponents are a mathematical way to express repeated multiplication of the same number. They consist of a base and an exponent. The base is the number that will be multiplied, and the exponent tells us how many times we will multiply the base by itself. For example, in \(3^2\), 3 is the base, and 2 is the exponent.
When we read \(3^2\), it means \(3 imes 3\). If the base was 4 and the exponent was 3, it would be \(4^3 = 4 imes 4 imes 4\).

• The formal notation is \(b^n\), where \(b\) is the base and \(n\) is the exponent.
• An exponent of 2 is often called "squared," while an exponent of 3 is "cubed."

Working with exponents follows specific rules, such as the product of powers, power of a power, and power of a product rules. However, with simple square calculations, understanding the concept is often enough.

Handling Negative Numbers

Negative numbers are numbers less than zero. They are represented with a minus sign (-) before the number. Understanding how to compute with negative numbers is crucial when dealing with expressions involving them.
In the expression \((-3)^2\), the negative sign before the number is part of the base. It is important to recognize this because it affects how we square the number.

• When you multiply two negative numbers, the result is positive.
• Thus, \((-3) imes (-3)\) equals \(9\), because multiplying two negatives results in a positive outcome.

Practicing with negative numbers enhances your confidence in accurately computing various mathematical expressions.

The Basics of Multiplication

Multiplication is a fundamental mathematical operation that involves adding a number to itself a specified number of times. For example, multiplying \(3 imes 2\) means adding 3 twice, which equals 6.
This operation is key in evaluating expressions like \( (-3)^2 \).

• It is depicted by the times symbol \(\times\).
• In the order of operations, multiplication has higher precedence than addition and subtraction.
• Understanding multiplication's rules are essential, especially in more complex problems that involve different operations.

Grasping these basic concepts of multiplication allows mathematicians to handle more advanced mathematical calculations with ease.