Exponent Rules Cheat Sheet

Every exponent law with a worked example, plus the powers, squares, and cubes you look up most — on one printable page. Free, no signup, no ads on the printout.

🖨 Print this page Download the PDF

The 9 laws of exponents

Product rulea^m · aⁿ = a^m+n2³ · 2² = 2⁵ = 32

Quotient rulea^m ÷ aⁿ = a^m−n3⁵ ÷ 3² = 3³ = 27

Power of a power(a^m)ⁿ = a^m·n(2²)³ = 2⁶ = 64

Power of a product(a·b)ⁿ = aⁿ · bⁿ(2·3)² = 4 · 9 = 36

Power of a quotient(a/b)ⁿ = aⁿ / bⁿ(3/4)² = 9/16

Zero exponenta⁰ = 1 (a ≠ 0)7⁰ = 1

Negative exponenta⁻ⁿ = 1 / aⁿ2⁻³ = 1/8 = 0.125

Fractional exponenta^1/n = ⁿ√a9^1/2 = √9 = 3

Rational exponenta^m/n = ⁿ√(a^m)8^2/3 = (³√8)² = 4

Watch out — the two classic mistakes:

(a + b)² ≠ a² + b² — (2 + 3)² = 25, but 2² + 3² = 13. Exponents don’t distribute over addition.
−3² = −9, but (−3)² = 9. Without parentheses, the exponent applies before the minus sign.

Powers of 2
2¹	2
2²	4
2³	8
2⁴	16
2⁵	32
2⁶	64
2⁷	128
2⁸	256
2⁹	512
2¹⁰	1,024
2¹¹	2,048
2¹²	4,096
2¹³	8,192
2¹⁴	16,384
2¹⁵	32,768
2¹⁶	65,536
2¹⁷	131,072
2¹⁸	262,144
2¹⁹	524,288
2²⁰	1,048,576

Powers of 10
10^-6	0.000001	millionth
10^-3	0.001	thousandth
10^-2	0.01	hundredth
10^-1	0.1	tenth
10⁰	1	one
10¹	10	ten
10²	100	hundred
10³	1,000	thousand
10⁶	1,000,000	million
10⁹	1,000,000,000	billion
10¹²	1,000,000,000,000	trillion

Squares
1²	1
2²	4
3²	9
4²	16
5²	25
6²	36
7²	49
8²	64
9²	81
10²	100
11²	121
12²	144
13²	169
14²	196
15²	225
16²	256
17²	289
18²	324
19²	361
20²	400

Cubes
1³	1
2³	8
3³	27
4³	64
5³	125
6³	216
7³	343
8³	512
9³	729
10³	1,000

Bonus — the Rule of 72: divide 72 by a growth rate to estimate doubling time. Money at 8% doubles in about 72 ÷ 8 = 9 years. It works because growth is an exponent: (1.08)⁹ ≈ 2.

Free printable from exponent-calculator.com — the fast, accurate exponent calculator. Teachers and students: copy, print, and share freely.

How to use this cheat sheet

Print it (the print version strips the website chrome and fits on one page), keep the PDF on your phone, or bookmark this page. When a problem needs an exact answer — like 2⁴² or 1.07²⁰ — use the exponent calculator or the future value calculator to get the full number instantly. If you’re learning the rules for the first time, the step-by-step guide on how to calculate exponents walks through each one.

Want the full-length tables? See the complete powers of 2 table (to 2⁶⁴), squares to 100² and cubes to 20³, and the compound growth factor table (1% to 15%).

Did you know?

“Exponent” is Latin for “placed out.” The word was coined by Michael Stifel in 1544, from exponere — the little number literally “set out” above the line.
Descartes invented the superscript. Writing x³ instead of xxx comes from René Descartes’ La Géométrie in 1637 — before that, mathematicians wrote powers out in words.
Why we type 2^10. The caret symbol became “to the power of” because early keyboards couldn’t type superscripts — programming languages adopted ^ and it stuck.
The 0⁰ controversy. Most calculators say 0⁰ = 1, but mathematicians call it an indeterminate form whose value depends on context — a 200-year-old argument that’s still not fully settled.
“Squared” and “cubed” are geometry. n² is literally the area of a square with side n, and n³ the volume of a cube — the names survive from ancient Greek geometry.
A number too big to write. A googolplex is 10 to the power of a googol — writing its digits would need more space than the observable universe contains.
Towers of exponents. Beyond powers lies tetration — 2↑↑4 means 2^{2^2²} = 2 to the 65,536th power, a number with nearly 20,000 digits.
The strangest minimum. The function x^x doesn’t keep shrinking as x gets small — it bottoms out at x = 1/e ≈ 0.368, then turns back up. Even exponents have a favorite number.

Exponent Rules Cheat Sheet

The 9 laws of exponents

How to use this cheat sheet

Sources & Further Reading