FundamentalsTopic #2 of 32

Exponents and Radicals

Laws of exponents, radical expressions, and simplification rules.

Laws of Exponents

Rule	Formula
Product Rule	$a^m \cdot a^n = a^{m+n}$
Quotient Rule	$\frac{a^m}{a^n} = a^{m-n}$
Power Rule	$(a^m)^n = a^{mn}$
Product to Power	$(ab)^n = a^n b^n$
Quotient to Power	$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$
Zero Exponent	$a^0 = 1$ $(a \neq 0)$
Negative Exponent	$a^{-n} = \frac{1}{a^n}$
Fractional Exponent	$a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$

Properties of Radicals

Basic Properties

Property	Formula
Definition	$\sqrt[n]{a} = a^{1/n}$
Product	$\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}$
Quotient	$\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}$
Power	$\sqrt[n]{a^m} = a^{m/n}$
Nested Radicals	$\sqrt[m]{\sqrt[n]{a}} = \sqrt[mn]{a}$

Simplifying Radicals

$\sqrt{a^2 b} = a\sqrt{b} \quad \text{(where } a \geq 0\text{)}$

$\sqrt[3]{a^3 b} = a \cdot \sqrt[3]{b}$

$\sqrt[n]{a^n} = |a| \quad \text{(when n is even)}$

$\sqrt[n]{a^n} = a \quad \text{(when n is odd)}$

Rationalizing Denominators

Single Term Denominator

$\frac{a}{\sqrt{b}} = \frac{a\sqrt{b}}{b}$

$\frac{a}{\sqrt[n]{b}} = \frac{a \cdot \sqrt[n]{b^{n-1}}}{b}$

Binomial Denominator (Conjugates)

$\frac{a}{\sqrt{b} + \sqrt{c}} = \frac{a(\sqrt{b} - \sqrt{c})}{b - c}$

$\frac{a}{\sqrt{b} - \sqrt{c}} = \frac{a(\sqrt{b} + \sqrt{c})}{b - c}$

Common Square Roots

Value	Result
$\sqrt{1}$	$1$
$\sqrt{4}$	$2$
$\sqrt{9}$	$3$
$\sqrt{16}$	$4$
$\sqrt{25}$	$5$
$\sqrt{36}$	$6$
$\sqrt{49}$	$7$
$\sqrt{64}$	$8$
$\sqrt{81}$	$9$
$\sqrt{100}$	$10$

Special Values

Expression	Value
$\sqrt{2}$	$\approx 1.414$
$\sqrt{3}$	$\approx 1.732$
$\sqrt{5}$	$\approx 2.236$
$\sqrt[3]{2}$	$\approx 1.260$

Arithmetic Operations

Back to all topics