Strategy #11: Know how to Apply Exponent Rules

Category: Advanced Math

The core rules:

• xᵃ · xᵇ = xᵃ⁺ᵇ (same base, multiply → add exponents)

• xᵃ / xᵇ = xᵃ⁻ᵇ (same base, divide → subtract exponents)

• (xᵃ)ᵇ = xᵃᵇ (power of a power → multiply exponents)

• x⁰ = 1 (any nonzero base to the zero power = 1)

• x⁻ᵃ = 1/xᵃ (negative exponent → reciprocal)

Why it matters: Exponent problems appear in almost every SAT, and mistakes here are usually due to confusing the rules under time pressure.

Practice tip: Write out the rule explicitly before applying it. Don't try to do it in your head until you've drilled each rule hundreds of times.

Try this simple example (solution is below):

Which expression is equivalent to (m⁴q⁴z⁻¹)(mq⁵z³), where m, q, and z are positive?
A) m⁴q²⁰z⁻³
B) m⁵q⁹z²
C) m⁶q⁸z⁻¹
D) m²⁰q¹²z⁻²

Solution:

Multiply the powers with the same bases using the multiplication property of exponents: add the exponents on matching bases.
(m⁴q⁴z⁻¹)(mq⁵z³) = m⁴⁺¹q⁴⁺⁵z⁻¹⁺³ = m⁵q⁹z²
The answer is B.