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The 7 Exponent Rules You Actually Need to Know

April 12, 2026 · Math

Exponents show up everywhere — compound interest, scientific notation, computer science, population growth. If you're comfortable with the core rules, a lot of math suddenly gets easier. Not just tolerable. Easier.

Let's get into it.

The quick refresher

An exponent tells you how many times to multiply a number by itself. That's it. 24 means 2 × 2 × 2 × 2, which is 16. The base is 2, the exponent is 4.

an = a × a × a × ... (n times)

The 7 rules

1. Product rule — same base, add exponents

When you multiply two numbers with the same base, just add the exponents.

am × an = am+n

So 32 × 34 = 36 = 729. Don't multiply the bases — that's a classic mistake.

2. Quotient rule — same base, subtract exponents

Same idea for division. Subtract the bottom exponent from the top one.

am / an = am-n

Example: 57 / 53 = 54 = 625.

3. Power of a power — multiply the exponents

When you raise a power to another power, multiply them together.

(am)n = am×n

(23)4 = 212 = 4,096. This one comes up a lot in algebra, so it's worth memorizing.

4. Power of a product — distribute the exponent

If a product is inside parentheses with an exponent outside, the exponent applies to each factor individually.

(ab)n = an × bn

Handy shortcut: (2x)3 = 23 × x3 = 8x3.

5. Power of a quotient — same deal for division

(a/b)n = an / bn

Works the same way. Just don't confuse this with the quotient rule — here you're distributing the exponent, not subtracting.

6. Zero exponent — everything becomes 1

This one feels wrong the first time you see it. Anything (except 0) raised to the power of 0 equals 1.

a0 = 1

70 = 1. 1000 = 1. x0 = 1. The reasoning comes from the quotient rule — an / an = an-n = a0, and any number divided by itself is 1.

7. Negative exponent — flip it

A negative exponent means take the reciprocal. Move the base to the other side of the fraction line.

a-n = 1 / an

So 2-3 = 1 / 23 = 1/8 = 0.125. It's not a negative number — the exponent just flips the fraction.

The mistakes everyone makes (at least once)

  • (a + b)2 does NOT equal a2 + b2. You have to FOIL it out: (a + b)2 = a2 + 2ab + b2. This is probably the single most common algebra error.
  • Don't add bases with different numbers. 23 + 32 is not 55. Calculate each one separately (8 + 9 = 17), then add.
  • Don't multiply bases with different numbers either. 23 × 32 is not 65. It's 8 × 9 = 72.

For quick calculations without second-guessing yourself, our exponent calculator handles all of this instantly.

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