Simplifying Radicals and Radical Operations

Perfect Squares

$1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, ...$

Simplifying Square Roots

Find the largest perfect square factor:

$\sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2}$ $\sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2}$ $\sqrt{48} = \sqrt{16 \cdot 3} = 4\sqrt{3}$

Product Property

$\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$

Quotient Property

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

$\sqrt{\frac{25}{9}} = \frac{\sqrt{25}}{\sqrt{9}} = \frac{5}{3}$

Adding and Subtracting Radicals

Only combine like radicals (same radicand):

$3\sqrt{5} + 7\sqrt{5} = 10\sqrt{5}$ $6\sqrt{2} - 2\sqrt{2} = 4\sqrt{2}$

Cannot combine: $3\sqrt{2} + 4\sqrt{3}$ (unlike radicals)

Multiplying Radicals

$\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$

$\sqrt{3} \cdot \sqrt{12} = \sqrt{36} = 6$

Rationalizing the Denominator

Remove radicals from the denominator:

$\frac{5}{\sqrt{3}} = \frac{5}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{5\sqrt{3}}{3}$

Solving Radical Equations

$\sqrt{x + 3} = 5$ $x + 3 = 25$ $x = 22$

Always check: $\sqrt{22 + 3} = \sqrt{25} = 5$ ✓

Watch out: Always check solutions to radical equations — squaring both sides can introduce extraneous solutions!