Problem 68 - Intermediate Algebra

Question

Write a sentence explaining, in your own words, how to evaluate the expression \((-8)^{2}\). Also write a sentence explaining how to evaluate \(-8^{2}\).

Step-by-Step Solution

Answer

\((-8)^2 = 64\); \(-8^2 = -64\).

Step 1: Identifying the Expression

First, let's identify the two expressions we need to evaluate. The first expression is \((-8)^2\), whereas the second one is \(-8^2\). The key difference is the presence or absence of parentheses around the negative number.

Step 2: Evaluating the First Expression: \((-8)^2\)

To evaluate \((-8)^2\), recognize that the parentheses indicate that the base number is -8. Thus, we need to square -8, calculated as follows: \((-8) \times (-8) = 64\). The result is positive because multiplying two negative numbers results in a positive number.

Step 3: Evaluating the Second Expression: \(-8^2\)

In the second expression \(-8^2\), the absence of parentheses means that only 8 is the base, and the negative sign is taken into account after calculating the square of 8. This means:First calculate \(8^2 = 64\), and then apply the negative sign: \(-64\). So, \(-8^2 = -64\).

Step 4: Conclusion: Comparing the Expressions

When evaluating \((-8)^2\), both the negative sign and the number 8 are included as the base, resulting in a positive 64. However, for \(-8^2\), only the number 8 is squared, and the negative sign is outside, resulting in -64.

Key Concepts

Negative Numbers

Negative numbers can often feel a bit tricky, especially when you are performing operations like exponentiation. Simply put, a negative number is any number less than zero. For example, -8 is less than zero, so it is a negative number. When you multiply two negative numbers together, the product becomes positive. This is crucial when dealing with expressions such as exponentiation because the base might be negative.

• Example: \(-8 imes -8 = 64\) because two negatives multiply to make a positive.

However, remember that if a negative number is raised to an even power, the result is positive. Conversely, raising a negative number to an odd power will yield a negative result.
This property is important when squaring negative numbers, as it affects the outcome dramatically depending on how the negative sign is handled.

Parentheses in Expressions

Parentheses play a crucial role in mathematics, especially in expressions involving negative numbers. In mathematics, parentheses are used to group numbers and operations together to clarify how to perform calculations. With exponentiation and negative numbers, parentheses dictate whether the negative sign is part of the base or not.
Consider the expression \( (-8)^2 \). Here, the parentheses indicate that -8 is the base of the exponentiation. You will square -8, resulting in a positive 64 because \( (-8) imes (-8) = 64 \).
On the other hand, with the expression \(-8^2\), the absence of parentheses means only 8 is the base, and the negative sign does not participate in the squaring. You square 8 to get 64, then apply the negative sign, resulting in -64.

• In expressions like \( (-a)^n \), the negative number is included in the base.
• In expressions like \(-a^n\), the base is positive, and the negative sign is applied after computation.

Hence, the presence or absence of parentheses can define the way expressions are calculated.

Order of Operations

The order of operations is a fundamental rule in mathematics dictating the sequence in which operations should be performed to accurately compute expressions. Known by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), it helps avoid confusion and ensures consistency in mathematical calculations.
With expressions involving both parentheses and exponents, the order of operations becomes vital. Consider \(-8^2\), where the absence of parentheses implies that exponentiation is performed before the application of the negative sign:

• First, calculate \(8^2 = 64\).
• Then, apply the negative sign to get \(-64\).

This progression is vital to obtain the correct result. However, if parentheses are around the negative number, like in \((-8)^2\), the negative sign is part of the base:

• First, ensure the entire base, \((-8)\), is squared.
• The multiplication calculation \((-8) imes (-8)\) equals \(64\).

Following the order of operations ensures clarity and accuracy in expressions comprising multiple mathematical elements.