Adding and Subtracting Radicals

Combining like radicals

Operations with Radicals

Like radicals have the same radicand (number under the radical).

Like radicals: $3\sqrt{5}$ and $7\sqrt{5}$ NOT like radicals: $3\sqrt{5}$ and $3\sqrt{2}$

You can only add or subtract like radicals. Combine the coefficients.

Example 1: $3\sqrt{5} + 7\sqrt{5} = (3 + 7)\sqrt{5} = 10\sqrt{5}$

Example 2: $8\sqrt{3} - 2\sqrt{3} = (8 - 2)\sqrt{3} = 6\sqrt{3}$

Sometimes you need to simplify radicals before you can combine them.

Example: $\sqrt{12} + \sqrt{27}$ $= 2\sqrt{3} + 3\sqrt{3}$ $= 5\sqrt{3}$

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

Example: $\sqrt{2} \cdot \sqrt{8} = \sqrt{16} = 4$

Add: $5\sqrt{7} + 3\sqrt{7}$

💡 Show Solution

These are like radicals (both have $\sqrt{7}$ ).

Combine the coefficients: $5\sqrt{7} + 3\sqrt{7} = (5 + 3)\sqrt{7} = 8\sqrt{7}$

Answer: $8\sqrt{7}$

Simplify: $\sqrt{18} + \sqrt{8}$

💡 Show Solution

Step 1: Simplify each radical $\sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}$ $\sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2}$

Step 2: Now they are like radicals, so add them $3\sqrt{2} + 2\sqrt{2} = 5\sqrt{2}$

Answer: $5\sqrt{2}$

Multiply: $\sqrt{6} \cdot \sqrt{10}$

💡 Show Solution

Use the product property: $\sqrt{6} \cdot \sqrt{10} = \sqrt{6 \cdot 10} = \sqrt{60}$

Now simplify: $\sqrt{60} = \sqrt{4 \cdot 15} = 2\sqrt{15}$

Answer: $2\sqrt{15}$

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