Power of a Power Calculator
Calculate (b^m)^n using the power of a power rule: multiply the exponents
Calculate Power of a Power
The base number
The inner exponent
The outer exponent
Formula
(b^m)^n = b^(m × n)
(0^0)^0 = 0^(0 × 0) = 0^0
Calculation Results
Exponent Rules
Power of a Power
(a^m)^n = a^(m×n)
Multiply the exponents
Product Rule
a^m × a^n = a^(m+n)
Add the exponents
Quotient Rule
a^m ÷ a^n = a^(m-n)
Subtract the exponents
Special Cases
Zero Exponent
a^0 = 1 (where a ≠ 0)
Negative Exponent
a^(-n) = 1/a^n
Fractional Exponent
a^(m/n) = ⁿ√(a^m)
Example Calculation
Problem
Calculate (2³)⁴
Solution
(2³)⁴ = 2^(3×4)
= 2^12
= 4,096
Alternative: 2³ = 8, then 8⁴ = 4,096
Understanding the Power of a Power Rule
What is the Power of a Power Rule?
The power of a power rule is a fundamental exponent law that states when you raise a power to another power, you multiply the exponents together while keeping the same base. This rule simplifies complex exponential expressions.
Mathematical Formula
(b^m)^n = b^(m × n)
where b is the base, m is the first power, n is the second power
Why Does This Work?
When you have (b^m)^n, you're essentially multiplying b^m by itself n times. Using the product rule of exponents (a^x × a^y = a^(x+y)), you add the exponent m a total of n times, which equals m × n.
Step-by-Step Example
Example: (3²)⁴
Step 1: Identify base = 3, m = 2, n = 4
Step 2: Apply rule: 3^(2×4) = 3^8
Step 3: Calculate: 3^8 = 6,561
Alternative Method
Step 1: Calculate inner: 3² = 9
Step 2: Calculate outer: 9⁴ = 6,561
Result: Same answer! ✓
With Negative Exponents
Example: (2⁻³)² = 2^(-3×2) = 2⁻⁶
Result: 1/2⁶ = 1/64
Important Notes
- • The base remains unchanged
- • Only multiply exponents, don't add them
- • Works with negative and fractional exponents
- • Be careful with negative bases and non-integer exponents