Problem 17 - Algebra A Combined Function

Question

Evaluate \((-4)^{2}\)

Step-by-Step Solution

Answer

The value of (-4)^{2} is 16.

Step 1: Identify the Base and Exponent

The problem presents us with the expression (-4)^{2}. Here, '(-4)' is the base and '2' is the exponent. The base is negative since it's enclosed in parentheses, indicating that the negative sign applies to the base as a whole.

Step 2: Apply the Exponent to the Base

To solve (-4)^{2}, we apply the exponent to the base by multiplying the base by itself. Since the exponent is 2, we multiply (-4) by (-4). In mathematical terms, this is written as (-4) imes (-4).

Step 3: Multiply the Negative Numbers

Multiply the two negative numbers: (-4) imes (-4) = 16. Remember that multiplying two negative numbers results in a positive number.

Key Concepts

Negative Numbers

Negative numbers can sometimes be tricky, especially when dealing with powers. They are numbers less than zero and are represented by a minus sign (-). Understanding how they interact with other numbers, particularly during multiplication and exponentiation, is key to solving problems efficiently.

When you see a negative base within parentheses, like (-4), this indicates that the entire number, including its sign, will be affected by any operations you perform. This distinction is crucial, especially in exponentiation, as it determines the result of the operation.

It's essential to remember that multiplying negative numbers follows specific rules:

• Two negative numbers multiplied together result in a positive number.
• If only one of the numbers in a multiplication is negative, the product will be negative.

Recognizing these rules helps simplify complex expressions and avoid common mistakes.

Multiplication

Multiplication is a fundamental arithmetic operation that is in essence repetitive addition. When multiplying, you take one number and add it to itself a certain number of times as specified by another number. For example, multiplying 3 by 4 (3 x 4) essentially means adding 3 together four times: 3 + 3 + 3 + 3 = 12.

When dealing with negative numbers in multiplication, remember to apply the rules that dictate the result's sign. Multiplying two negative numbers, as in our exercise with (-4) x (-4), results in a positive product.

In the context of exponents, multiplying a number by itself a certain number of times is exactly what an exponent represents. Whether the base number is positive or negative can significantly impact the final result, as highlighted by the problem we've solved.

Base and Exponent

The notion of base and exponent is central to understanding powers and how they operate. The base is the number that will be multiplied by itself. The exponent, a small number written to the right and slightly above the base, tells you how many times to multiply the base.

In the expression (-4)^{2}, the base is -4, and the exponent is 2. This means you take -4 and multiply it by itself once (as an exponent of 2 involves one multiplication operation of the base with itself).

An important point is when the base number is negative, the sign is included in the calculation only if enclosed in parentheses. Without parentheses, the sign may not apply to the exponentiation directly. In our original exercise, since -4 is enclosed in parentheses, the negative sign is part of the base. Thus, (-4)^{2} must be calculated with the negative base in consideration for accurate results. When properly calculated, (-4 x -4) equals 16, showing the power of utilizing both the base and exponent properly.