Multiplying Negative Exponents Calculator

This calculator helps you multiply expressions with negative exponents. Learn the rules and see step-by-step examples of how to handle negative exponents in multiplication.

How to Use This Calculator

Enter the base and exponent values for each term you want to multiply. The calculator will show you the step-by-step solution and the final result.

Tip: Remember that negative exponents indicate reciprocals. For example, \( a^{-n} = \frac{1}{a^n} \).

Rules for Multiplying Negative Exponents

When multiplying terms with exponents, follow these key rules:

1. Multiply the bases: \( a^m \times a^n = a^{m+n} \)

2. Add the exponents when the bases are the same

3. For negative exponents: \( a^{-m} \times a^{-n} = a^{-(m+n)} \)

Special Cases

When multiplying terms with different bases, keep them separate unless they can be combined
Negative exponents become positive when multiplied by their positive counterparts
Zero exponents result in 1 when multiplied with any term

Worked Examples

Example 1: Same Base

Calculate \( 2^{-3} \times 2^{-4} \)

Solution:

1. Apply the multiplication rule: \( 2^{-3} \times 2^{-4} = 2^{-3-4} = 2^{-7} \)

2. Final result: \( \frac{1}{128} \)

Example 2: Different Bases

Calculate \( 3^{-2} \times 5^{-3} \)

Solution:

1. Since bases are different, keep them separate: \( 3^{-2} \times 5^{-3} \)

2. Convert to positive exponents: \( \frac{1}{3^2} \times \frac{1}{5^3} = \frac{1}{9 \times 125} = \frac{1}{1125} \)

Frequently Asked Questions

Can I multiply terms with different negative exponents?

Yes, you can multiply terms with different negative exponents. If the bases are the same, add the exponents. If the bases are different, keep them separate unless they can be combined.

What happens when I multiply a negative exponent by its positive counterpart?

The negative and positive exponents cancel each other out, resulting in a positive exponent. For example, \( a^{-n} \times a^n = a^{0} = 1 \).

How do I handle multiplication with zero exponents?

Any term with a zero exponent equals 1, so multiplying by 1 doesn't change the product. For example, \( a^{-n} \times a^{0} = a^{-n} \times 1 = a^{-n} \).