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Chapter 4: Problem 11

Evaluate each exponential expression without using a calculator. $$-2^{0}$$

Short Answer

Expert verified
The expression evaluates to -1.

Step by step solution

01

- Understand the Properties of Exponents

First, recall the properties of exponents. One key property is that any nonzero number raised to the power of 0 is 1. This can be expressed as: \[ a^0 = 1 \] for any nonzero number \(a\).
02

- Apply the Zero Exponent Rule

Apply this rule to the given expression. Here, the base is -2 and the exponent is 0. According to the zero exponent rule: \[ (-2)^0 = 1 \]
03

- Evaluate the Expression

Finally, since we have determined that \( (-2)^0 = 1 \), we find: \[-2^0 = -1 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Properties of Exponents
Before diving into evaluating exponential expressions, it's crucial to understand the properties of exponents. Exponential expressions follow specific rules that simplify complex calculations. One essential property is the zero exponent rule. Another is that multiplying same bases adds their exponents: \[a^m * a^n = a^{m+n}\]. Also, raising a power to another power multiplies exponents: \[(a^m)^n = a^{m*n}\]. These properties are pivotal for simplifying and solving exponential expressions.
Grasping the Zero Exponent Rule
The zero exponent rule is simple yet powerful. It states that any nonzero number raised to the zero power equals 1. This can be surprising at first, but the reasoning lies in the consistent application of exponent rules. For example: \[a^m * a^n = a^{m+n}\]. If we set \(m\) to any number and \(n\) to zero, we get \[a^m * a^0 = a^{m+0} = a^m\], meaning \(a^0 = 1\). Always remember this rule when you see any number or variable raised to the power of zero.
Evaluating Exponential Expressions
When tasked with evaluating exponential expressions, follow a systematic approach.
  • First, identify and apply the relevant exponent rules.
  • Second, simplify the expression step-by-step.
  • Third, perform calculations carefully to avoid mistakes.
For instance, in the expression \(-2^0\), recognize the base is -2 and the exponent is 0. According to the zero exponent rule \((-2)^0 = 1\). Because the exponent only affects the base, the expression simplifies to: \[-2^0 = -1\]. It's that simple!

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Most popular questions from this chapter

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