Simplifying Radicals

Simplifying square roots and radical expressions

Square Root Basics

The square root of a number is a value that, when multiplied by itself, gives the original number.

$\sqrt{25} = 5 \text{ because } 5^2 = 25$

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

Example: $\sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3}$

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

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

Example: $\sqrt{\frac{9}{16}} = \frac{\sqrt{9}}{\sqrt{16}} = \frac{3}{4}$

Simplify: $\sqrt{36}$

💡 Show Solution

Find what number squared equals 36:

$6^2 = 36$

Therefore: $\sqrt{36} = 6$

Answer: $6$

Simplify: $\sqrt{50}$

💡 Show Solution

Step 1: Find the largest perfect square factor of 50 $50 = 25 \cdot 2$

Step 2: Use the product property $\sqrt{50} = \sqrt{25 \cdot 2}$ $= \sqrt{25} \cdot \sqrt{2}$ $= 5\sqrt{2}$

Answer: $5\sqrt{2}$

Simplify: $\sqrt{72}$

💡 Show Solution

Step 1: Find the largest perfect square factor of 72 $72 = 36 \cdot 2$

Step 2: Apply the product property $\sqrt{72} = \sqrt{36 \cdot 2}$ $= \sqrt{36} \cdot \sqrt{2}$ $= 6\sqrt{2}$

Answer: $6\sqrt{2}$

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