Exponentiation

Exponentiation

Exponentiation is an operation which takes two numbers: the base $b$ and the exponent $n$. It produces the power $b^n$. For positive integers, $b^n$ means multiplying $b$ by itself $n$ times.

Identities

$b^n\cdot b^m=b^{n+m}$ $(b^n{)}^m=b^{nm}$ $b^0=1(b\ne 0)$ ^18c142 $b^{-n}=\frac{1}{b^n}(b\ne 0)$ $b^{n/m}=\sqrt[m]{b^n}$ $1^n=1$ $(-1{)}^n=\{\begin{array}{ll}1,& n\text{ even}\\ -1,& n\text{ odd}\end{array}$ $0^n=0,n>0$

The edge cases matter: $0^0$ is context-dependent and is usually left undefined in elementary algebra, though it may be assigned $1$ in combinatorics or power-series conventions.

Power function

A power function has variable base and fixed exponent:

$$f_{power}(x)=x^n.$$

^e477f8 It is inverted by a Root when the domain is chosen carefully. For example, $x^2$ is not one-to-one on all real numbers, so its inverse is usually taken on $x\ge 0$.

Exponential function

An exponential function has fixed base and variable exponent:

$$f_{exponential}(x)=b^x.$$

^5f7806 It is inverted by a Logarithm. The usual real-valued assumptions are $b>0$ and $b\ne 1$.

Calculus and modelling

For power functions, the power rule gives

$$\frac{d}{dx}{x}^n=n{x}^{n-1}.$$

For exponentials,

$$\frac{d}{dx}{b}^x=b^x\ln b.$$

The natural exponential $e^x$ is special because $\ln e=1$, so it differentiates to itself. That self-reproducing derivative is why exponentials solve proportional-rate problems such as Growth and decay.

Common pitfalls

• Multiplying powers with different bases as if $a^n{b}^m=(ab{)}^{n+m}$; this is generally false.
• Forgetting parentheses: $-2^2=-(2^2)=-4$, but $(-2{)}^2=4$.
• Applying real exponent laws blindly to negative bases and fractional powers.

See also Exponents, Logarithms, and Roots, Root, and Logarithm.