EQN-03-02

Laws of Exponents

The five exponent rules that turn scary-looking powers into one-line answers.

~5 min read•Difficulty 2/5

1. Product : a^{m} \cdot a^{n} = a^{m + n}

2. Quotient : a^{m} / a^{n} = a^{m - n}

3. Power : (a^{m})^{n} = a^{m \cdot n}

4. Zero : a^{0} = 1 (a \neq = 0)

5. Negative : a^{- n} = 1/ a^{n}

The product and quotient rules only work when the base is the same. Rewrite numbers as powers of a common base to use them.

2^{5} \cdot 2^{3} = 2^{8}

8 \cdot 4 = 2^{3} \cdot 2^{2} = 2^{5} = 32

$(a^{m})^{n}$ multiplies the exponents, but $a^{m} \cdot a^{n}$ adds them. They are different.

(3^{2})^{3} = 3^{6} = 729

3^{2} \cdot 3^{3} = 3^{5} = 243

a = a^{1/2}, 3 a = a^{1/3}

a \cdot b = a \cdot b

Worked examples

Example 1 (difficulty 1/5)

Simplify

: 5^{7} / 5^{4}

◎ Predict the answer first:

Example 2 (difficulty 2/5)

2^{x} = 32,

what is x ?

◎ Predict the answer first:

Memorize powers of 2 up to $2^{10} = 1024$ and squares up to 20² — they appear constantly.
When bases differ, try to rewrite as powers of a single small base (2, 3, or 5).
A negative exponent never makes a number negative — it makes a reciprocal.

Multiplying exponents when multiplying powers

: 2^{3} \cdot 2^{4} = 2^{12}

Why: The "power of a power" rule leaks into the product rule.

Fix: Multiplying same-base powers ADDS exponents; only

(a^{m})^{n}

multiplies them.

Thinking

a^{0} = 0

Why: "Zero of anything" feels like zero.

Fix: Any nonzero base to the power 0 equals 1.