Problem 7 - College Algebra

Question

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

Step-by-Step Solution

Answer

The expression \(-3^{0}\) evaluates to 1.

Step 1: Understanding Exponential Notation

Exponential notation involves raising a number (known as the base) to a certain power (known as the exponent). Here, the base is -3 and the exponent is 0.

Step 2: Apply the Power of Zero Rule

In mathematics, any number (except zero) raised to the power of zero is 1. This is known as the 'Power of Zero Rule'.

Step 3: Compute the Expression

Following the 'Power of Zero Rule', the expression \(-3^{0}\) evaluates to 1.

Key Concepts

Power of Zero Rule

The 'power of zero rule' is a fundamental concept in mathematics that simplifies many computations involving exponents. It states that any non-zero number raised to the zeroth power is equal to one, irrespective of the value of the base number. Thus, when you see an expression like \( b^{0} \), where \( b \) is any non-zero number, you can confidently say that it evaluates to 1. This rule is particularly useful because it helps maintain consistency in the properties of exponents. For example, consider the exponential expression \( (-3)^{0} \). Applying the power of zero rule would mean that this expression is equal to 1. The rule emphasizes the fact that raising a number to the power of zero doesn't lead to a huge value or a tiny fraction, but rather, it simplifies everything down to the number one.

Exponential Notation

Exponential notation is a concise way to represent repeated multiplication of the same factor. In this notation, a number, known as the 'base', is multiplied by itself a certain number of times indicated by the 'exponent'. This is written as \( b^{n} \), where \( b \) is the base and \( n \) is the exponent. For example, the expression \( 3^{4} \) means that the number 3 is multiplied by itself 4 times: \( 3 \times 3 \times 3 \times 3 \). When trying to understand an exponential expression, it’s important first to identify the base and the exponent. Once you do that, you can follow the specific rules related to evaluating exponents, such as the power of zero rule. Exponential notation becomes particularly handy for expressing very large numbers, such as the speed of light (approximately \( 3^{8} \) meters per second), or very small numbers, such as the size of an atom (approximately \( 10^{-10} \) meters).

Evaluating Exponents

Evaluating exponents requires a clear understanding of both the 'power of zero rule' and the 'exponential notation'. Once you recognize the base and the exponent, you can determine the value of the expression by multiplying the base by itself, the number of times indicated by the exponent. There are special rules when the exponent is zero, negative or a fraction. When the exponent is zero, as we learned, the result is always one for any non-zero base. If the exponent is negative, it means we are dealing with a reciprocal (for instance, \( b^{-n} \) is equal to \( 1/{b^{n}} \)). When dealing with fractional exponents, the numerator tells you the power, and the denominator tells you the root (so \( b^{1/n} \) represents the nth root of \( b \)). With these rules in mind, evaluating \( (-3)^{0} \) involves recognizing that the exponent is zero and therefore the result is 1, regardless of the negative base.