Power of a Power, Power of a Product
Simple Explanation
Raising a power to another power multiplies the exponents: (aᵐ)ⁿ = aᵐⁿ. Raising a whole product or quotient to a power distributes that power to every factor: (ab)ⁿ = aⁿbⁿ and (a/b)ⁿ = aⁿ/bⁿ.
Why Do We Need It?
Expressions with nested powers, or powers of whole products, show up constantly in algebra — these rules let you expand or simplify them in one step instead of writing out every factor.
Formula
Power-of-a-Power and Power-of-a-Product Rules
(aᵐ)ⁿ = aᵐⁿ and (ab)ⁿ = aⁿbⁿ and (a/b)ⁿ = aⁿ/bⁿ
Raising a power to another power multiplies the exponents; raising a product or quotient to a power distributes that power to every factor.
- a, b
- — the bases being raised to a power
- m, n
- — the exponents
When to use it: Whenever an expression already raised to a power is itself raised to another power, or when a whole product/quotient is raised to a power.
Worked Example
Simplify a power of a power and a power of a product
Simplify (x³)⁴ and (2y)³.
Why Does This Work?
(aᵐ)ⁿ means aᵐ multiplied by itself n times, which by the product rule adds m to itself n times — the same as multiplying m by n. A product raised to a power is that same product multiplied by itself n times, and multiplication can be reordered so each factor collects its own n copies.
Real-Life Example
Scaling a 3D model
A cube-shaped storage container is scaled up so every edge (originally length s) becomes 3 times as long.
Its volume, originally s³, becomes (3s)³ = 3³s³ = 27s³ — the power-of-a-product rule shows the volume grows 27 times, not just 3 times.
Practice
Simplify (m⁵)².
EasyEvaluate (3x)² when x = 1 (just find the numeric coefficient of x²).
MediumCommon mistake
Adding the exponents instead of multiplying them in (aᵐ)ⁿ — that mistake confuses this rule with the product rule for aᵐ · aⁿ, which is a completely different situation.
Quick Review
- (aᵐ)ⁿ = aᵐⁿ — a power of a power multiplies the exponents.
- (ab)ⁿ = aⁿbⁿ and (a/b)ⁿ = aⁿ/bⁿ — the power distributes to every factor.
- Do not confuse power-of-a-power (multiply exponents) with the product rule (add exponents).