Problem 14 - Algebra and Trigonometry

Question

\(7-28\) Evaluate each expression. $$ (-3)^{2} $$

Step-by-Step Solution

Answer

The value 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 tells us that the base is multiplied by itself once. It's important to observe that the negative sign is included in the base.

Step 2: Apply the Exponent to the Base

Apply the exponent 2 to the base -3, which means:\[(-3) \times (-3)\].

Step 3: Multiply the Numbers

Now multiply the numbers:\[-3 \times -3 = 9.\]Recall that multiplying two negative numbers results in a positive product.

Key Concepts

Negative numbers

Negative numbers are values less than zero, represented with a minus sign.
They are the opposite of positive numbers and are often used to represent losses, debts, or positions below zero.
In mathematics, understanding how negative numbers interact with operations like multiplication is essential.

When multiplying negative numbers, remember a few key points:

• Multiplying two negative numbers results in a positive number.
  This is because the two negative signs cancel each other out.
• Multiplying a positive number by a negative number results in a negative number.
  This indicates a change in direction on the number line.
• The absolute value of a negative number is the same as its positive counterpart.
  For instance, the absolute value of -3 is 3.

Understanding these properties helps in evaluating expressions accurately.
Keep in mind that enclosing a negative number in parentheses helps avoid confusion when exponentiation is involved, as in this exercise.

Multiplication

Multiplication is a mathematical operation used to find the total of one number added repeatedly.
It's a faster way of adding the same number several times.
For example, a multiplied by b is the same as adding a to itself b times.

In the context of exponentiation, multiplication plays a crucial role, especially in expressions like \((-3)^2\).
Here, the number -3 is multiplied by itself, leading to:

• First multiplication: -3 (the base) times -3 = 9, since multiplying two negative numbers gives a positive product.

This method effectively demonstrates how multiplication intersects with negative numbers and exponents.
Remember, practicing multiplication regularly with different signs reinforces the concept and aids in understanding complex mathematical scenarios.

Base and exponent

The base and exponent are components of an expression that define repeated multiplication.
The base is the number that is multiplied, and the exponent indicates how many times the base is multiplied by itself.

For example, in the expression \((-3)^2\),

• The base is -3. This means -3 is the number being multiplied.
• The exponent is 2. This means the base is used two times in multiplication.

When applying the exponent to the base, it is essential to include the entire base, including any negative sign, inside the parentheses.
This ensures the exponent applies to the whole value and not just the numeric part.
Understanding base and exponent concepts helps simplify expressions with powers and solves problems accurately.Furthermore, knowing the distinction between expressions like \((-3)^2\) and -3^2 can be critical, as the latter is often interpreted without parentheses, leading to a different outcome.