Repeated multiplication made short
← Back to Math TopicsAn exponent (or power) tells you how many times to multiply a number by itself. The big number is the base and the small raised number is the exponent.
We read \(2^4\) as “two to the fourth power.” The base is \(2\) and the exponent is \(4\).
\(5^2 = 5 \times 5 = 25\)
An exponent of 2 is called “squared.”
\(2^3 = 2 \times 2 \times 2 = 8\)
An exponent of 3 is called “cubed.”
\(7^1 = 7\)
Any number to the power 1 is itself.
\(9^0 = 1\)
Any number (except 0) to the power 0 equals 1.
These shortcut rules make working with powers much faster:
\(a^m \times a^n = a^{m+n}\)
Same base? Add the exponents. \(2^3 \times 2^2 = 2^5 = 32\)
\(a^m \div a^n = a^{m-n}\)
Same base? Subtract the exponents. \(5^6 \div 5^4 = 5^2 = 25\)
\((a^m)^n = a^{m \times n}\)
Multiply the exponents. \((3^2)^3 = 3^6 = 729\)
\((ab)^n = a^n b^n\)
Share the power. \((2 \times 5)^2 = 4 \times 25 = 100\)
\(a^{-n} = \dfrac{1}{a^n}\)
Flip it. \(2^{-3} = \dfrac{1}{8}\)
\(a^{1/n} = \sqrt[n]{a}\)
A root in disguise. \(9^{1/2} = \sqrt{9} = 3\)
A root asks the opposite question: “what number, multiplied by itself, gives this?” The square root \(\sqrt{\;}\) undoes squaring, and the cube root \(\sqrt[3]{\;}\) undoes cubing.
| n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|
| \(n^2\) | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 |
| \(n^3\) | 1 | 8 | 27 | 64 | 125 | 216 | 343 | 512 | 729 | 1000 |