Problem 9 - College Algebra

Question

Evaluate each exponential expression. $$ -3^{0} $$

Step-by-Step Solution

Answer

The result of evaluating the expression \(-3^{0}\) is 1.

Step 1: Identify the Base and the Exponent

In the expression \(-3^{0}\), the base is \(-3\) and the exponent is 0.

Step 2: Apply the Zero Exponent Rule

Any non-zero number to the power of zero is always 1. Here, the base is \(-3\) (not zero), so \((-3)^{0} = 1\). It's important to note that the parenthesis are crucial. If we had \(-3^{0}\) without parenthesis, the negative sign wouldn't be included in the base, and the answer would be different.

Step 3: Write the Final Answer

So, \(-3^{0} = 1\).

Key Concepts

Zero Exponent Rule

One of the coolest concepts in mathematics is the Zero Exponent Rule. This simply says that any non-zero number raised to the power of zero equals one. In other words, this rule states:

• For any number or variable \(a\), if \(a eq 0\), then \(a^{0} = 1\).

This might feel a bit strange at first, as it seems like magically turning something into one. However, it has a solid mathematical foundation based on the pattern observed in the division of exponents. When dividing powers of the same base, you subtract the exponents, and this idea leads to the conclusion of the zero exponent. Next, let’s connect this to the concept of base and exponent to grasp this more closely.

Base and Exponent

The terms "base" and "exponent" are fundamental to understanding exponential expressions. When you see something like \(-3^{0}\), it’s expressed in the form \(a^{n}\). Here:

• \(a\) is the base - it's the number that is being multiplied.
• \(n\) is the exponent - it tells us how many times to multiply the base by itself.

In the expression \(-3^{0}\),

• The base is \(-3\).
• The exponent is 0.

So, you don't multiply negative three at all because you are elevating it to the zeroth power, indicating merely the base exists to one single instance. Understanding this helps in evaluating exponential expressions confidently.

Evaluating Negative Base

Evaluating a negative base in an exponential expression can sometimes generate confusion. It's crucial to recognize the importance of parentheses when dealing with negative bases. Let’s break it down:

• In \((-3)^{0}\), the entire base \(-3\) is raised to the power of zero.
• If there were no parentheses, as in \(-3^{0}\), only the 3 is raised to the zeroth power, and the result is negative: \(-(3^{0}) = -1\).

So when solving \((-3)^{0}\), ensure the negative sign is included inside the base when parentheses are present. This often overlooked detail can change the outcome of the evaluation completely, showing you the power of proper notation in mathematics. Remember, clear notation = clear understanding!