# Summary

Algebraic expressions can often be simplified and subsequently solved through the use of the exponent laws (also called laws of indices or power laws). These laws allow an equation to be manipulated into a form which provides enhanced readability or opens up potential simplifications and substitutions. This article provides a reference for these laws.

# Exponent Laws

Ref Formula
1

$\displaystyle (x^{a})^{b} = x^{ab}$

2

$\displaystyle x^{(a+b)} = x^{a} \times x^{b}$

3

$\displaystyle x^{a/b} = (x^{a})^{1/b}$

4

$\displaystyle x^{-a} = \frac{1}{x^a}$

5

$\displaystyle x^{a-b} = \frac{x^{a}}{x^{b}}$

6

$\displaystyle \left( \frac{x}{y} \right)^a = \frac{x^{a}}{y^{a}}$

7

$\displaystyle x^{1/2} = \sqrt{x}$

8

$\displaystyle x^{1/a} = \sqrt[a]{x}$

9

$\displaystyle \sqrt{x^2} = \left|x\right|$

10

$\displaystyle \sqrt{xy} = \sqrt{x}\sqrt{y}$

$\displaystyle \text{for } x \gt 0, y \gt 0$

11

$\displaystyle \sqrt[b]{x^a} = x^{a/b}$

$\displaystyle \text{For } x \gt 0$

12

$\displaystyle \sqrt[n]{\frac{1}{x}} = \frac{1}{\sqrt[n]{x}}$

13

$\displaystyle \left( \frac{x}{y} \right)^{-a} = \left( \frac{y}{x} \right)^{a}$