Power of a Power Calculator
Calculate (am)n by multiplying the exponents
How to Use This Calculator
Enter Base
Input the base number (a) that is raised to the first power, which is then raised to the second power.
Enter First Exponent
Input the exponent (m) for the inner power - the power to which the base is first raised.
Enter Second Exponent
Input the exponent (n) for the outer power - the power to which the first result is raised.
Get Result
Click "Calculate" to see the result. The calculator multiplies the exponents: (a^m)^n = a^(m×n).
Formula
(am)n = am×n
Where:
- a = base number
- m = first exponent (inner power)
- n = second exponent (outer power)
Example 1: (2³)²
(2³)² = 23×2 = 2⁶ = 64
Verification: (8)² = 64 ✓
Example 2: (5²)³
(5²)³ = 52×3 = 5⁶ = 15,625
Verification: (25)³ = 15,625 ✓
Example 3: (10²)⁴
(10²)⁴ = 102×4 = 10⁸ = 100,000,000
Verification: (100)⁴ = 100,000,000 ✓
About Power of a Power Calculator
The Power of a Power Calculator helps you calculate (am)n, which equals am×n. When you raise a power to another power, you multiply the exponents. This is one of the fundamental rules of exponents and is essential for simplifying complex exponential expressions.
When to Use This Calculator
- Algebra: Simplify expressions like (x²)³ when solving equations
- Exponential Functions: Simplify expressions before graphing or analyzing
- Calculus: Simplify expressions before differentiating or integrating
- Scientific Notation: Simplify complex powers in scientific calculations
- Mathematical Simplification: Reduce complex exponential expressions to simpler forms
Why Use Our Calculator?
- ✅ Instant Results: Get accurate power of a power calculations immediately
- ✅ Clear Explanation: See the step-by-step application of the exponent rule
- ✅ Educational: Learn the power of a power rule 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)³ = 64, you can simplify: (2ˣ)³ = 23x = 64 = 2⁶, so 3x = 6, which means x = 2.
Calculus: When differentiating or integrating expressions like (x²)⁵, first simplify: (x²)⁵ = x¹⁰, making the derivative or integral easier.
Exponential Functions: Simplify expressions like (e2t)³ = e6t for easier analysis.
Tips for Best Results
- Remember: (a^m)^n = a^(m×n) - exponents are multiplied
- This rule simplifies nested powers significantly
- Works with any real number exponents (positive, negative, or fractional)
- You can apply this rule multiple times: ((a^m)^n)^p = a^(m×n×p)
- The result has the same base with the product of the exponents
Frequently Asked Questions
What happens when I raise a power to another power?
When raising a power to another power, you multiply the exponents: (am)n = am×n. For example, (2³)² = 2⁶ = 64.
Is (a^m)^n the same as a^(m^n)?
No! (a^m)^n = a^(m×n) multiplies the exponents, while a^(m^n) raises m to the nth power first. For example, (2³)² = 2⁶ = 64, but 2^(3²) = 2⁹ = 512.
Can I apply this rule with more than two levels?
Yes! For nested powers, multiply all the exponents: ((a^m)^n)^p = a^(m×n×p). For example, ((2²)³)² = 2^(2×3×2) = 2¹² = 4,096.
What if one exponent is negative?
The rule still applies! For example, (2²)⁻³ = 22×(-3) = 2⁻⁶ = 1/2⁶ = 1/64. Multiplying by a negative exponent makes the result negative.
Why is this rule important?
This rule is essential for simplifying complex exponential expressions. Without it, you'd have to calculate each power separately, which is much more work. It's one of the fundamental exponent rules used throughout mathematics.