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An exponent in a power function is the number of times the base value is multiplied by itself.

## Exponents

Where any value base $a$ is being "raised to the power" $y$, the value of $y$ is said to be the exponent. It is the number of times base $a$ is to be multiplied by itself; that is:

$$a^y = \displaystyle\prod_{k=1}^{y} = a\cdot a\cdot\ldots\cdot a$$

Any nonzero value with a zero exponent is equal to $±1$:

$$a^0 = 1$$ if $$a$$ is positive, $$-1$$ if $$a$$ is negative, or undetermined if $$a = 0$$.

Values with negative exponents are reciprocals of the base raised to the power of the associated positive exponent:

$$a^{-1}=\frac{1}{a^1}$$,  $$a^{-2}=\frac{1}{a^2}$$,  etc.

Values with fractional exponents are nth-root functions in the form of $$x^{\frac{m}{n}} = \sqrt[n]{x^m}$$, therefore $$x^{\frac{1}{n}} = \sqrt[n]{x}$$.

## Nth-Roots

An nth-root function is the inverse of a power function; that is, if $$x^n=y$$, then $$\sqrt[n]{y}=x$$. For example, if we square the number $$5$$, (that is, we perform the operation $$x=5^2$$), we get a result of $$5\cdot 5=25$$. Therefore, the square root of $$25$$ is $$5$$. Mathematically, we would write this as ($$\sqrt{25}=5$$). Likewise, if we cube the number $$5$$, we get $$125$$, so ($$\sqrt{125}=5$$).

## Exponential Identities

This list of identities shows equivalent forms of many power functions and will help when working problems involving exponents and roots.

$$x^m = \frac{1}{x^{-m}}$$
$$x^{-m} = \frac{1}{x^m}$$
$$x^{\frac{1}{n}} = \sqrt[n]{x}$$
$$x^{-\frac{1}{n}} = \frac{1}{\sqrt[n]{x}} = \sqrt[n]{\frac{1}{x}}$$
$$x^{\frac{m}{n}} = \sqrt[n]{x^m} = (\sqrt[n]{x})^m$$
$$x^{-\frac{m}{n}} = \frac{1}{\sqrt[n]{x^m}} = \sqrt[n]{\frac{1}{x^m}} = \sqrt[n]{x^{-m}}$$
$$x^{m+n} = x^m\cdot x^n$$
$$x^{m+\frac{1}{n}} = x^m\cdot \sqrt[n]{x}$$
$$x^{m-n} = \frac{x^m}{x^n}$$
$$x^{mn} = (x^m)^n$$
$$x^ny^n = (xy)^n$$
$$\frac{x^n}{y^n} = (\frac{x}{y})^n$$
$$\sqrt[n]{pq} = \sqrt[n]{p}\cdot \sqrt[n]{q}$$
$$\sqrt[n]{\frac{p}{q}} =\frac{\sqrt[n]{p}}{\sqrt[n]{q}}$$
$$\sqrt[n]{p^nq} = p\sqrt[n]{q}$$
$$\frac{\sqrt[n]{x}}{x} = \frac{1}{\sqrt[n]{x}}$$
$$x^n = a^{n\log_a{x}}$$
$$\sqrt[n]{x} = a^{\frac{\log_a{x}}{n}}$$