Problem 14 - Introductory and Intermediate Algebra for College Students 4th

Question

Evaluate each exponential expression. $$-8^{2}$$

Step-by-Step Solution

Answer

The result of evaluating the expression \( -8^{2} \) is \( 64 \).

Step 1: Understand the rules of exponentiation

In this expression, \( -8 \) is the base and \( 2 \) is the exponent. This means that the base, \( -8 \), should be multiplied by itself \( 2 \) times.

Step 2: Multiply the base by itself

After understanding the meaning of the exponent notation, carry out the multiplication. Multiply \( -8 \) by \( -8 \) to get \( 64 \).

Step 3: Write down the result

After carrying out the multiplication, write down the result. The expression \( -8^{2} \) simplifies to \( 64 \).

Key Concepts

Exponential Expression

When we come across the term "exponential expression," it might sound daunting at first, but it's easier than it seems. An exponential expression involves a number raised to a power. More formally, it is written in a format like this: the base, followed by an exponent, which looks like a tiny raised number. This setup tells us how many times to use the base in a multiplication.
Exponential expressions are popular in mathematics because they offer a compact way to represent repeated multiplication. Instead of writing a number several times, we let the exponent handle that information.

• In the expression \(-8^{2}\), "-8" is our base, and "2" is the exponent.
• The expression is telling us to multiply "-8" by itself once.
• The format simplifies long multiplications into a more straightforward notation.

Understanding and evaluating exponential expressions is a crucial skill in mathematics because it comes up often, whether in basic algebra or advanced calculus tasks.

Base and Exponent

In any exponential expression, two key parts make up the notation: the base and the exponent. Each plays a vital role in forming the complete expression, like pieces of a puzzle.
- **Base**: This is the number that you will multiply by itself.- **Exponent**: This value represents how many times you need to multiply the base.
In the example of \(-8^{2}\):

• The base is "-8". It's the number we focus on multiplying.
• The exponent is "2". This tells us "multiply -8 by itself two times."

The interaction between the base and the exponent defines the magnitude and sign of the result. The base carries the sign and value, while the exponent adds the repetitive action of multiplication. This is why signs and operations inside an exponential expression can dramatically affect the final outcome.

Multiplication

Multiplication in the context of exponentiation is essentially a repeated operation. Exponential notation, such as in \(-8^{2}\), simplifies this process by using the exponent to eliminate the need for repetitive writing.
Here’s how you operate using exponentiation:

• You start with the base, which is "-8" in our example.
• Then, according to the exponent "2", you multiply "-8" by itself: \((-8) \times (-8) = 64\).

The product of multiplying two negative numbers is positive, which is why \(-8^{2}\) results in 64. Negative bases can often trick students, so it's essential to remember:
- An even exponent results in a positive number when the base is negative.This demonstrates how crucial understanding the basics of multiplication is in solving and making sense of exponential expressions.