Problem 6 - Algebra and Trigonometry

Question

Evaluate each expression. $$ -6^{0} $$

Step-by-Step Solution

Answer

The value of \(-6^0\) is \(-1\).

Step 1: Understanding the problem

We need to evaluate the expression \(-6^0\). This involves understanding the rules of exponents and how to apply them to this expression.

Step 2: Apply the Zero Exponent Rule

The Zero Exponent Rule states that any non-zero number raised to the power of zero equals 1. However, it applies directly to the base if there's no parentheses. Here, \(6^0\) equals 1, but the negative sign is not inside the base which means it stays as -1.

Step 3: Combine the Negative Sign with the Answer

Since the base affected by the exponent is only 6 (not -6), we initially find \(6^0 = 1\). Then, apply the negative sign to the result. This gives us \(-6^0 = -1\).

Key Concepts

Zero Exponent Rule

Have you ever wondered what happens when you raise a number to the power of zero? Unlike what one might initially think, raising a number to zero does not mean it equals zero! In the world of exponents, the zero exponent rule is a handy rule: any non-zero number raised to the power of zero equals one. For example, \(5^0 = 1\), \( (100)^0 = 1 \), or \( (-3.14)^0 = 1 \). This concept also applies to more complex expressions and is an essential part of simplifying problems involving exponents. When applying the zero exponent rule, it's critical to recognize that it only affects the base. That means whatever number or expression inside the exponentiation is replaced by one, while terms outside of the exponent remain unaffected at this step. This rule is one of the fundamental properties of exponents, making calculations much simpler when it is applied correctly.

Negative Sign Handling

When dealing with expressions involving exponents and negative signs, it's important to know exactly where the negative sign falls. This can make a significant difference in the outcome. For instance, compare

• \(-a^b\)
• \((-a)^b\)

In the first case, the negative sign is not part of the base, meaning the exponent only affects \(a\), and the negative sign is applied afterward. So for \(-a^b\), the expression is evaluated by considering \(a^b\) first, and then the result is multiplied by \(-1\). In the original exercise of \(-6^0\), this concept is crucial. The negative sign in front of the base 6 is considered separately from the exponentiation, leading to identifying the base as \(6\), and therefore applying the zero exponent results in \(6^0 = 1\). The negative sign then makes the result \(-1\). It’s vital to carefully observe the position of negative signs in expressions with exponents to avoid miscalculations.

Order of Operations

The order of operations is a fundamental concept in mathematics, ensuring clarity and consistency in solving mathematical expressions. Commonly remembered by the acronym PEMDAS, it stands for:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

This order dictates the sequence in which operations should be performed. When evaluating expressions, it's essential to follow these guidelines to arrive at the correct answer. In the case of \(-6^0\), following the order of operations means addressing the exponent first. The expression \(6^0\) is evaluated while ignoring the initial negative sign, simplifying to 1. Only after handling the exponents do we consider multiplication and division, if any, with the negative sign then applied. This systematic approach ensures accuracy and prevents confusion, making complex expressions more manageable. Always remember, executing operations in the right order can be the difference between the correct result and an incorrect one.