Roots

A root of x is a number that, when raised to a certain power (the index), equals the radicand.

The default root is 2: $\sqrt{x} = \sqrt[2]{x}$

Square Root

The square root is a number that, when multiplied by itself, equals the radicand.

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

Cubic Root

The cubic root is a number that, when raised to the power of three, equals the radicand.

$\sqrt[3]{1} = 1~~~\sqrt[3]{8} = 2\\ \sqrt[3]{27} = 3~~~\sqrt[3]{64} = 4\\ \sqrt[3]{125} = 5~~~\sqrt[3]{216} = 6\\ \sqrt[3]{343} = 7~~~\sqrt[3]{512} = 8\\ \sqrt[3]{729} = 9~~~\sqrt[3]{1000} = 10$

Simplifying Radical Expressions

Simplifying Radicals - Best Explanation (youtube)
Learn how to simplify a radical expression with variables and a number (youtube)

Radical Rules

• Fractional Exponent Rule:
  $\sqrt[n]{x^m} = x^{\frac{m}{n}}$
• Product Rule:
  $\sqrt{xy} = \sqrt{x} \times \sqrt{y}$
• Power Rule 1:
  $(\sqrt{x})^y = \sqrt{x^y}$
• Power Rule 2:
  $\sqrt[n]{x^n} = x$
• Quotinent Rule:
  $\sqrt{{\frac{x}{y}}} = \frac{\sqrt{x}}{\sqrt{y}}$