Power of a Power – Multiplying Exponents

When a power is itself raised to another power – like (x²)³ – the two exponents combine by multiplying, not adding: (x^a)^b = x^ab. This makes sense once you unpack it: (x²)³ means “x², multiplied by itself 3 times,” which is x² × x² × x² – and by the product law from a couple of pages ago, that's x²⁺²⁺² = x⁶, exactly matching 2 × 3 = 6.

The Power of a Power Law

(x^a)^b = x^ab – multiply the two exponents together.

Simplify (4²)⁴ as a single power of 4.

Multiply the exponents: 2 × 4 = 8. So (4²)⁴ = 4⁸.

Checking with Real Numbers

Compute (2²)³ directly, then check it matches 2⁶.

(2²)³ = 4³ = 64. And 2⁶ = 64. They match! 64.

Real-Life Application

Algebra: simplifying expressions where a bracketed power is raised again.
Compound growth: repeatedly compounding a growth factor over several periods.
Computer science: analysing nested loops or repeated doubling processes.

Key Takeaways

Power of a power: (x^a)^b = x^ab – multiply the exponents.
This follows directly from repeatedly applying the product law.
Zero exponents, covered next, explain what happens when the quotient law produces an exponent of 0.

Practice: Power of a Power

Power of a Power