Problem 7 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

-9

Step 1: Understand the Expression

The given expression is \[-3^{2}\]. This expression means the square of \(3\) is being multiplied by \(-1\). It's important to note that the negative sign is not included in the squaring.

Step 2: Square the Number

Calculate the square of \(3\). \[3^{2} = 9\].

Step 3: Apply the Negative Sign

Now, apply the negative sign from the original expression. \(-1 \times 9 = -9\).

Step 4: Conclude the Evaluation

The result of evaluating the expression \(-3^{2}\) is \(-9\).

Key Concepts

Understanding Exponents

Exponents are a mathematical way of indicating that a number, known as the base, is to be multiplied by itself a specific number of times. For example, in the expression \(3^2\), the number 3 is the base, and the exponent is 2. You multiply the base by itself: \(3 \times 3\), resulting in 9. Exponents make it easier to write and calculate large numbers quickly.

• The exponent tells you how many times to use the base in a multiplication.
• They provide a shorthand method of expressing repeated multiplication.
• Exponents are crucial in both simple and complex mathematical calculations.

When using exponents, remember that they only affect the number directly next to them, not any negative sign that might precede it without a parenthesis. Therefore, -3 squared should be written in mathematics as \(-(3^2)\).

Grasping the Order of Operations

The Order of Operations is a set of rules that ensure mathematicians evaluate expressions consistently. This is often remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). Following this order ensures that everyone arrives at the same answer when working out a given problem.

• First, evaluate expressions inside parentheses.
• Second, calculate exponents.
• Then, perform multiplication and division, working from left to right.
• Finally, do addition and subtraction, again from left to right.

In the case of \(-3^2\), we follow the order of operations by first squaring 3 to get 9, then applying the negative sign. This ensures we are consistent in our approach to solving any algebraic expression.

Working with Arithmetic Expressions

Arithmetic expressions are combinations of numbers, operators, and, sometimes, parentheses. Solving these involves following the order of operations to simplify the expression into a single number or the simplest form possible. Arithmetic expressions are the foundation of more complex algebraic activities and require a good understanding of how basic mathematical operations interact.

• Operators include addition, subtraction, multiplication, and division.
• Expressions might also contain exponents and nested operations inside parentheses.
• The goal is to evaluate the expression accurately by applying arithmetic rules.

For example, the expression \(-3^2\) involves recognizing which parts to compute first based on established mathematical conventions. This foundational knowledge assists in the seamless transition to more advanced math concepts.