Problem 52 - Intermediate Algebra

Question

Evaluate each expression. See Example \(6 .\) $$ -8^{2} $$

Step-by-Step Solution

Answer

The expression evaluates to -64.

Step 1: Evaluate the Power

First, evaluate the power in the expression. The expression is \[-8^{2}\]. Here, only 8 is being squared, not the negative sign. Therefore, calculate: \[8^2 = 8 \times 8 = 64.\]

Step 2: Apply the Negative Sign

Now that the exponent has been evaluated, apply the negative sign to the result found in Step 1: \[-8^2 = -(8^2) = -64.\] This indicates that the negative sign applies to the whole result of the power calculation.

Key Concepts

Order of operations

Understanding the order of operations is crucial when evaluating algebraic expressions, especially those involving exponents. The order of operations refers to the rules that dictate the sequence in which you should solve parts of a math problem. This sequence ensures that everyone evaluates expressions the same way and achieves consistent results.

An easy way to remember the order of operations is by using the acronym PEMDAS:

• Parentheses – solve expressions inside parentheses first.
• Exponents – next, calculate powers and roots.
• Multiplication and Division – perform these operations from left to right.
• Addition and Subtraction – finally, perform these operations from left to right.

In the example we've looked at, \(-8^2\), we see the importance of applying the order correctly. We first evaluate the exponent, as indicated in PEMDAS, only after considering it separately from the negative sign.

Negative numbers in algebra

Negative numbers can be a stumbling block for many students. When they appear in algebraic expressions, it's important to understand how they interact with other numbers and operations, such as exponents.

When you see a negative number with an exponent, like \(-8^2\), it's important to realize that the negative sign is not part of the base of the exponent unless it is specifically included in parentheses (e.g., \((-8)^2\)). In our example, the negative sign is applied after squaring, rather than to the base itself.

Dealing with negative numbers involves straightforward rules:

• For multiplication or division, a negative times a positive yields a negative.
• A negative times a negative results in a positive.

By applying these rules consistently, you can avoid common mistakes with negative numbers in problems involving powers.

Power evaluation

Power evaluation involves calculating the value of a number raised to an exponent. This process is straightforward, but requires careful attention, especially in expressions involving other operations like subtraction or division.

Raising a number to a power means multiplying the number by itself a certain number of times. For instance, in \(8^2\), the calculation involves multiplying the number 8 by itself, resulting in 64. It's important to square the number or evaluate the power before applying any other operations, such as a negative sign in our example, \(-8^2\).

Remember these tips for power evaluation:

• Square and other power calculations come after parentheses but before multiplication, division, addition, and subtraction.
• Be careful with negative numbers. As discussed earlier, the negative sign in \(-8^2\) comes after calculating the power.
• Use a calculator or write out the calculation if you're unsure; this minimizes mistakes and builds confidence in your ability to handle exponents accurately.

In summary, thoroughly understanding power evaluation helps ensure you follow the correct order of operations and deal accurately with negative numbers in algebra.