Multiplying Exponents Calculator
Multiply powers with the same base by adding 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 powers being multiplied.
Enter First Exponent
Input the exponent (m) for the first power in the multiplication.
Enter Second Exponent
Input the exponent (n) for the second power in the multiplication.
Get Result
Click "Calculate" to see the result. The calculator adds the exponents: a^m × a^n = a^(m+n).
Formula
am × an = am+n
Where:
- a = base (same for both powers)
- m = first exponent
- n = second exponent
Example 1: 2³ × 2²
2³ × 2² = 23+2 = 2⁵ = 32
Verification: 8 × 4 = 32 ✓
Example 2: 5² × 5³
5² × 5³ = 52+3 = 5⁵ = 3,125
Verification: 25 × 125 = 3,125 ✓
Example 3: 10² × 10⁵
10² × 10⁵ = 102+5 = 10⁷ = 10,000,000
Verification: 100 × 100,000 = 10,000,000 ✓
About Multiplying Exponents Calculator
The Multiplying Exponents Calculator helps you multiply powers (exponentials) that share the same base. When you multiply two powers with the same base, you add 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: Multiply numbers in scientific notation (e.g., 10² × 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 multiplication of exponents immediately
- ✅ Clear Explanation: See the step-by-step application of the exponent rule
- ✅ Educational: Learn the product rule of exponents through examples
- ✅ 100% Free: No registration or payment required
- ✅ Accurate: Precise mathematical calculations with high precision
- ✅ Verification: Shows both the simplified form and the calculated value
Common Applications
Algebra: When solving equations like 2ˣ × 2³ = 256, you can simplify: 2x+3 = 256 = 2⁸, so x + 3 = 8, which means x = 5.
Scientific Notation: When multiplying very large numbers, (3 × 10²) × (2 × 10⁵) = (3 × 2) × 102+5 = 6 × 10⁷.
Exponential Growth: In modeling, you might need to simplify expressions like e2t × e3t = e2t+3t = e5t.
Tips for Best Results
- Remember: The bases must be the same to use this rule
- When multiplying: a^m × a^n = a^(m+n) - exponents are added
- This rule only works when multiplying powers with the same base
- You can multiply more than two powers: a^m × a^n × a^p = a^(m+n+p)
- The result will have the same base with the sum of the exponents
Frequently Asked Questions
What happens when I multiply exponents with the same base?
When multiplying powers with the same base, you add the exponents: am × an = am+n. For example, 2³ × 2² = 2⁵ = 32.
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 2³ × 3², you must calculate each power separately: 8 × 9 = 72.
Can I multiply more than two powers?
Yes! When multiplying multiple powers with the same base, add all the exponents: am × an × ap = am+n+p. For example, 2² × 2³ × 2⁴ = 2⁹.
What if one exponent is negative?
The rule still applies! For example, 5² × 5⁻³ = 52+(-3) = 5⁻¹ = 1/5 = 0.2. Adding negative exponents is like subtracting.
Is this the same as multiplying the bases?
No! This rule only applies when multiplying powers with the same base. For (am) × (bn) with different bases, you cannot add the exponents.