Dividing Exponents Calculator
Divide powers with the same base by subtracting their exponents
How to Use This Calculator
Enter the Base
Input the base number (a) that both exponents share. This is the same number in both the numerator and denominator.
Enter Numerator Exponent
Input the exponent (m) for the numerator, which is the power in the top part of the division.
Enter Denominator Exponent
Input the exponent (n) for the denominator, which is the power in the bottom part of the division.
Get Result
Click "Calculate" to see the result. The calculator subtracts the exponents: a^m / a^n = a^(m-n).
Formula
am ÷ an = am-n
Where:
- a = base (same for both powers)
- m = exponent in the numerator
- n = exponent in the denominator
Example 1: 2⁵ ÷ 2³
2⁵ ÷ 2³ = 25-3 = 2² = 4
Verification: 32 ÷ 8 = 4 ✓
Example 2: 10⁸ ÷ 10⁵
10⁸ ÷ 10⁵ = 108-5 = 10³ = 1,000
Verification: 100,000,000 ÷ 100,000 = 1,000 ✓
Example 3: 5⁴ ÷ 5⁶
5⁴ ÷ 5⁶ = 54-6 = 5⁻² = 1/5² = 1/25 = 0.04
When m < n, the result is a negative exponent
About Dividing Exponents Calculator
The Dividing Exponents Calculator helps you divide powers (exponentials) that share the same base. When you divide two powers with the same base, you subtract their exponents. This is one of the fundamental rules of exponents and is essential for simplifying algebraic expressions and solving exponential equations.
When to Use This Calculator
- Algebra: Simplify expressions like x⁵ / x² when solving equations
- Scientific Notation: Divide numbers in scientific notation (e.g., 10⁸ / 10⁵)
- Exponential Functions: Simplify expressions before graphing or analyzing
- Calculus: Simplify expressions before differentiating or integrating
- Mathematical Simplification: Reduce complex exponential expressions to simpler forms
Why Use Our Calculator?
- ✅ Instant Results: Get accurate division of exponents immediately
- ✅ Clear Explanation: See the step-by-step application of the exponent rule
- ✅ Handles Negative Exponents: Correctly calculates results when m < n
- ✅ Educational: Learn the quotient rule of exponents through examples
- ✅ 100% Free: No registration or payment required
- ✅ Accurate: Precise mathematical calculations with high precision
Common Applications
Algebra: When solving equations like 2ˣ / 2³ = 16, you can simplify: 2x-3 = 16 = 2⁴, so x - 3 = 4, which means x = 7.
Scientific Notation: When dividing very large numbers, (6 × 10⁸) / (2 × 10⁵) = (6/2) × 108-5 = 3 × 10³ = 3,000.
Exponential Growth/Decay: In modeling, you might need to simplify expressions like e3t / et = e3t-t = e2t.
Tips for Best Results
- Remember: The bases must be the same to use this rule
- When m > n, the result is positive: 2⁵ / 2² = 2³ = 8
- When m = n, the result is 1: 5⁴ / 5⁴ = 5⁰ = 1
- When m < n, the result has a negative exponent: 3² / 3⁵ = 3⁻³ = 1/27
- This rule only works when dividing powers with the same base
Frequently Asked Questions
What happens when I divide exponents with the same base?
When dividing powers with the same base, you subtract the exponents: am / an = am-n. For example, 2⁵ / 2³ = 2² = 4.
Can I use this rule if the bases are different?
No, this rule only applies when the bases are the same. For different bases like 2⁵ / 3², you must calculate each power separately: 32 / 9 ≈ 3.556.
What if the numerator exponent is smaller than the denominator exponent?
You'll get a negative exponent. For example, 5² / 5⁴ = 52-4 = 5⁻² = 1/5² = 1/25 = 0.04.
What if both exponents are the same?
When m = n, the result is always 1: am / an = am-m = a⁰ = 1. For example, 7⁵ / 7⁵ = 7⁰ = 1.
Is this the same as dividing the bases?
No! This rule only applies when dividing powers with the same base. For (am) / (bn) with different bases, you cannot subtract the exponents.