The Power Rule of Exponents: Mastering (a^m)^n = a^(m×n)
Master the power rule (a^m)^n = a^(m×n) with formulas, worked examples, and step-by-step solutions for nested exponent expressions.
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The power rule of exponents states that raising a power to another power multiplies the exponents: (a^m)^n = a^(m×n). This essential rule simplifies nested exponential expressions and is fundamental to algebra, scientific notation, and compound interest calculations.
Key Takeaway
When raising a power to another power, multiply the exponents: (a^m)^n = a^(m×n). This is different from the product rule which adds exponents.
1. The Power Rule Defined
This rule works because (a^m)^n means multiplying a^m by itself n times: a^m × a^m × ... × a^m (n times). By the product rule, this equals a^(m + m + ... + m) = a^(m×n).
SEO Professional Insight
The power rule is frequently confused with the product rule. Remember: product rule adds exponents (a^m × a^n = a^(m+n)), power rule multiplies them ((a^m)^n = a^(m×n)).
2. Worked Examples
Example 1: (2³)⁴
- Base: a = 2, inner exponent: m = 3, outer exponent: n = 4
- Apply power rule: (2³)⁴ = 2^(3×4) = 2¹²
- 2¹² = 4096
- Verify: 2³ = 8, 8⁴ = 4096 —
Example 2: Nested powers ((x²)³)⁴
- Multiply all exponents: x^(2×3×4) = x²⁴
- The rule extends to any number of nested powers
3. Comparison with Other Rules
| Rule | Formula | Operation |
|---|---|---|
| Product Rule | a^m × a^n = a^(m+n) | Add exponents |
| Power Rule | (a^m)^n = a^(m×n) | Multiply exponents |
| Quotient Rule | a^m / a^n = a^(m-n) | Subtract exponents |
4. Applications
- Scientific notation: (3 × 10²)⁴ = 81 × 10⁸ = 8.1 × 10⁹
- Compound interest: ((1+r)^n)^t = (1+r)^(nt)
- Roots as powers: (a^(1/n))^n = a
- Exponential equations: Solving for unknowns in nested expressions
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