Exponent Notation & Meaning

Base and exponent

In $2^{3}$, the $2$ is the base and the $3$ is the exponent. The exponent counts how many times the base is used as a factor: $2^{3} = 2 \cdot 2 \cdot 2 = 8$.

You read $2^{3}$ as "two to the third power." Two exponents have nicknames: a power of $2$ is called squared (because $s^{2}$ gives the area of a square) and a power of $3$ is called cubed (because $s^{3}$ gives the volume of a cube). So $7^{2}$ is "seven squared" and $5^{3}$ is "five cubed."

The mistake everyone makes once

An exponent does not mean multiply the base by the exponent. Compare: $2^{4} = 2 \cdot 2 \cdot 2 \cdot 2 = 16$, but $2 \cdot 4 = 8$. When you evaluate a power, write out the repeated factors first — it takes five seconds and it makes this mistake impossible.

Order matters too. Swapping the base and the exponent changes the value: $2^{6} = 64$ while $6^{2} = 36$. A bigger base does not automatically mean a bigger answer, because the exponent controls how many times you multiply.

Writing repeated multiplication as a power

Going the other direction is just counting. If the same factor repeats, that factor is the base and the count of copies is the exponent: $7 \cdot 7 \cdot 7 = 7^{3}$.

The factors have to be identical for this to work. $7 \cdot 7 \cdot 5$ is not a single power, because $5$ is not another copy of $7$ — you could only write it as $7^{2} \cdot 5$.

Worked examples

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Common questions