# Math Insight

### Basic rules for exponentiation

If $n$ is a positive integer and $x$ is any real number, then $x^n$ corresponds to repeated multiplication \begin{gather*} x^n = \underbrace{x \times x \times \cdots \times x}_{n \text{ times}}. \end{gather*} We can call this “$x$ raised to the power of $n$,” “$x$ to the power of $n$,” or simply “$x$ to the $n$.” Here, $x$ is the base and $n$ is the exponent or the power.

From this definition, we can deduce some basic rules that exponentiation must follow as well as some hand special cases that follow from the rules. In the process, we'll define exponentials $x^a$ for exponents $a$ that aren't positive integers.

The rules and special cases are summarized in the following table. Below, we give details for each one.

Rule or special caseFormulaExample
Product$x^ax^b = x^{a+b}$$2^22^3 = 2^5=32 Quotient\displaystyle \frac{x^a}{x^b} = x^{a-b}$$\displaystyle \frac{2^3}{2^2} = 2^1 =2$
Power of power$(x^a)^b = x^{ab}$$(2^3)^2 = 2^6=64 Power of a product(xy)^a = x^ay^a$$36=6^2=(2\cdotbadbreak 3)^2 = 2^2\cdotbadbreak 3^2=4 \cdotbadbreak 9=36$
Power of one$x^1=x$$2^1=2 Power of zerox^0=1$$2^0=1$
Power of negative one$\displaystyle x^{-1}=\frac{1}{x}$$\displaystyle 2^{-1}=\frac{1}{2} Change sign of exponents\displaystyle x^{-a} = \frac{1}{x^a}$$\displaystyle 2^{-3} = \frac{1}{2^3} = \frac{1}{8}$