Problem 21 - College Algebra with Modeling and Visualization

Question

Exercises \(17-34:\) Evaluate the expression by hand. Check your result with a calculator. $$ 5^{0} $$

Step-by-Step Solution

Answer

The expression \(5^0\) equals 1.

Step 1: Understanding the Power of Zero

In mathematics, any non-zero number raised to the power of zero is equal to 1. This is a standard rule that applies to exponents. Therefore, when we have the expression \(5^0\), we know by this rule that the answer must be 1.

Step 2: Applying the Rule

Now that we understand the rule, we can directly apply it to our expression. We have \(5^0\). According to the rule \(a^0 = 1\), where \(a\) is any non-zero number, we find that \(5^0 = 1\).

Step 3: Verification with a Calculator

To verify our answer, let's use a calculator to compute \(5^0\). When you enter 5 and raise it to the power of zero, the calculator should return 1, thus confirming our manual computation.

Key Concepts

Power of Zero

In the realm of exponents, the 'Power of Zero' is a fundamental concept that might seem paradoxical at first. This rule states that any non-zero number raised to the power of zero equals one. For example, in the expression \(5^0\), the result is 1. This might lead some to wonder why multiplying zero times seems to result in one.
This rule holds because of the pattern observed in exponentiation. For instance, as you decrease the exponent: \(2^3 = 8\), \(2^2 = 4\), \(2^1 = 2\), each step down (subtracting one from the exponent) divides the previous result by the base (2), making the pattern \(2^0 = 1\) logical. Since the base is not zero, this pattern applies universally, leading to the conclusion that the power of zero is always 1 for any non-zero base.

Mathematical Rules

Understanding mathematical rules, especially those related to exponents, is crucial in evaluating expressions accurately and effortlessly. The Power of Zero is just one of many rules. Others include:

• Product of Powers Rule: When multiplying two powers with the same base, you add the exponents: \(a^m \times a^n = a^{m+n}\).
• Quotient of Powers Rule: When dividing two powers with the same base, subtract the exponents: \(a^m / a^n = a^{m-n}\) (where \(a eq 0\)).
• Power of a Power Rule: A power raised to another power requires multiplying the exponents: \((a^m)^n = a^{m \times n}\).
• Negative Exponent Rule: A negative exponent represents a reciprocal: \(a^{-m} = 1/a^m\) (where \(a eq 0\)).

By getting familiar with these rules, you can easily evaluate various expressions and manipulate exponential terms with confidence.

Calculator Verification

Once you've applied the rules of mathematics by hand, using a calculator is an excellent way to verify your findings. Calculators are especially useful for avoiding human errors or miscalculations in complex problems. For our specific problem \(5^0\), you simply enter 5, then use the function to raise it to the power of zero.
When the calculator returns the result of 1, it provides a quick confirmation of the rule. While manual computation strengthens understanding, calculator verification ensures accuracy. This process is helpful, especially during exams or when solving complicated problems, by giving you peace of mind that your manual calculations are correct.