How to Calculate A Negative Bases with A Negative Exponents

Introduction

Calculating negative bases with negative exponents might seem confusing at first, but it follows a clear mathematical pattern. The key is understanding how negative signs interact with exponents. When a negative base is raised to a negative exponent, the result is positive.

This guide will walk you through the formula, provide step-by-step instructions, show worked examples, and explain common mistakes to avoid.

Basic Formula

The general formula for calculating a negative base with a negative exponent is:

For example, if you have \( -2^{-3} \), you first calculate \( 2^3 = 8 \), then take the reciprocal to get \( \frac{1}{8} \).

Step-by-Step Calculation

1. Identify the base and exponent. For example, \( -3^{-2} \).
2. Remove the negative sign from the exponent. Now you have \( 3^{-2} \).
3. Apply the exponent to the base. \( 3^2 = 9 \).
4. Take the reciprocal of the result. \( \frac{1}{9} \).
5. Apply the original negative sign to the final result. \( -\frac{1}{9} \).

Worked Examples

Example 1: \( -4^{-1} \)

1. Remove the negative exponent: \( 4^{-1} \).
2. Calculate \( 4^1 = 4 \).
3. Take the reciprocal: \( \frac{1}{4} \).
4. Apply the negative sign: \( -\frac{1}{4} \).

Final result: \( -\frac{1}{4} \).

Example 2: \( -5^{-2} \)

1. Remove the negative exponent: \( 5^{-2} \).
2. Calculate \( 5^2 = 25 \).
3. Take the reciprocal: \( \frac{1}{25} \).
4. Apply the negative sign: \( -\frac{1}{25} \).

Final result: \( -\frac{1}{25} \).

Common Mistakes

• Applying the negative sign to the base before calculating the exponent. This is incorrect because the negative sign is part of the base, not the exponent.
• Forgetting to take the reciprocal when dealing with negative exponents. Remember, \( a^{-n} = \frac{1}{a^n} \).
• Confusing negative bases with negative exponents. These are different concepts that require different approaches.