| Definition |
\(x^n=\overset{n\text{-times}}{\overbrace{x\cdot x\cdot x\cdots x}}\) |
\(n\in\mathbb{N}\) |
| Comparison |
\(x^m=x^n\iff m=n\) |
\(x\neq 0,\pm 1\); i.e., no special bases |
Same base
Different exponents* |
\(x^m\cdot x^n=x^{m+n}\) |
Multiplication: add exponents |
| \(x^m\div x^n=x^{m-n}\) |
Division: subtract exponents |
Same exponent
Different bases* |
\(x^m\cdot y^m=(x\cdot y)^m\) |
|
| \(x^m\div y^m=(x\div y)^m\) |
|
| Double exponent* |
\((x^m)^n=x^{mn}\) |
Exponents are multiplied |
| Negative exponents* |
\(x^{-1}=\dfrac{1}{x}\) |
\(x\neq0\)
Switch from numerator to denominator or vice vesa.
The exponent loses the negative.
The sign of the base is unchanged. |
| \(x^{-n}=\dfrac{1}{x^n}\) |
| Rational exponents† |
\(x^{\frac{1}{n}}=\sqrt[n]{x}\) |
\(n\in\mathbb{N};n\gt 1\)
Denominator becomes the index of the root. |
| \(x^{\frac{m}{n}}=\sqrt[n]{x^m}=\left(\sqrt[n]{x}\right)^m\) |
| Special exponents‡ |
\(x^1=x\) |
An absent exponent means an exponent of \(1\) |
| \(x^0=1\) |
\(x\neq 0\) |
| Special bases‡ |
\(0^n=0\) |
\(n\gt 0\) |
| \(1^n=1\) |
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| \((-1)^n=\begin{cases} \phantom{-}1;& n\text{ is even} \\ -1;& n\text{ is odd}\end{cases}\) |
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