Problem 9 - College Algebra

Question

Evaluate each exponential expression in Exercises 1–22. $$ -3^{0} $$

Step-by-Step Solution

Answer

The evaluated expression \(-3^0 = 1\).

Step 1: Identify the base and the exponent

In our problem, the number \(-3\) is the base and the exponent is 0.

Step 2: Apply the zero exponent rule

Any non-zero number to the power of zero equals 1. Therefore, \(-3^{0} = 1\).

Step 3: Write down the final result

The evaluated expression is 1.

Key Concepts

Understanding the Zero Exponent Rule

The zero exponent rule is a fundamental concept in mathematics that states any non-zero number raised to the power of zero is always equal to 1. It's like a magical shortcut! Whether you're dealing with big numbers, small numbers, or negative ones, the rule remains the same:

• For example, \(5^0 = 1\).
• Also, \((-3)^0 = 1\).
• Even larger numbers like \(1000^0 = 1\).

This is because the exponential function describes how many times to multiply the base by itself. When the exponent is zero, you're not multiplying the base at all, which results in 1.
Remember, this rule only applies to non-zero bases. Zero to the power of zero is a more complex topic and often treated as undefined.

Base and Exponent Identification

Understanding which part of a mathematical expression is the base and which is the exponent is crucial for evaluating expressions. The base is the main number that's being multiplied, and the exponent tells you how many times to multiply it by itself.
For example, in the expression \(3^4\):

• 3 is the base.
• 4 is the exponent.

In our specific problem of \(-3^0\):

• \(-3\) is the base, even though it's negative.
• 0 is the exponent.

Correctly identifying these parts allows you to apply the rules of exponents more effectively, ensuring accuracy in your calculations.

Evaluating Exponential Expressions

When evaluating exponential expressions, it's all about breaking down the expression into understandable parts and applying the correct rules. Here's a simple approach:
1. **Identify the Base and Exponent:** Know what numbers you're dealing with.
2. **Apply the Relevant Rule:** Each rule applies to specific situations. For the zero exponent rule, if the exponent is zero and the base is non-zero, your answer is 1.
3. **Final Calculation:** Use the rules to simplify the expression.

• For \(-3^0\), the base is \(-3\) and the exponent is 0. By the zero exponent rule, it evaluates to 1.
• Another example, \(8^0 = 1\), since the base is non-zero.

Practice these steps with different numbers to gain confidence in solving exponential equations efficiently.