Power of a Power Calculator

Calculate (a^m)^n using the power of a power rule with step-by-step verification.

Enter Values

Result

(a^m)^n
6561
Expression (3^2)^4
Simplified Exponent (m x n) 8
Equivalent Form 3^8
Verification (a^m) 9
Verification (a^m)^n 6561

Step-by-Step Solution

(3^2)^4 = 3^(2x4) = 3^8 = 6561

Understanding the Power of a Power Rule

The power of a power rule is one of the fundamental laws of exponents. It states that when you raise a power to another power, you multiply the exponents. In mathematical notation: (am)n = am x n. This rule simplifies complex exponential expressions and is used extensively in algebra, calculus, and scientific computations.

How the Rule Works

Consider the expression (am)n. This means you are taking am and multiplying it by itself n times. Since am means "a multiplied by itself m times," repeating this n times gives you a multiplied by itself m x n times total. This is why the exponents multiply.

Power of a Power

Multiply the exponents when raising a power to a power.

(a^m)^n = a^(m*n)

Power of a Product

Distribute the exponent to each factor in the product.

(ab)^n = a^n * b^n

Power of a Quotient

Distribute the exponent to numerator and denominator.

(a/b)^n = a^n / b^n

Product of Powers

Add exponents when multiplying same bases.

a^m * a^n = a^(m+n)

Examples and Applications

Example 1: Simple Integers

(23)2 = 23x2 = 26 = 64. Verification: 23 = 8, and 82 = 64.

Example 2: Negative Exponents

(5-2)3 = 5-2x3 = 5-6 = 1/15625. The rule works the same way with negative exponents.

Example 3: Fractional Exponents

(41/2)6 = 4(1/2)x6 = 43 = 64. Since 41/2 = 2, we can verify: 26 = 64.

Common Mistakes to Avoid

  • Do not add exponents instead of multiplying them: (am)n is NOT am+n
  • Do not confuse with am * an = am+n (product rule adds exponents)
  • Remember that the base stays the same; only the exponents change
  • For negative bases, pay attention to whether the final exponent is even or odd