Operations with Radicals

Adding radicals: simplify first, then combine like terms

You can only add/subtract radicals that have the same radicand (number under √). So simplify each radical first, then combine those with matching radicands — just like combining $3x + 2x = 5x$.

Step-by-step: $3\sqrt{8} + 2\sqrt{18} - \sqrt{50}$

Step 1 — Simplify each radical individually:

$\sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2}$

$\sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}$

$\sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2}$

Step 2 — Substitute back:

$3(2\sqrt{2}) + 2(3\sqrt{2}) - 1(5\sqrt{2})$

Step 3 — Multiply coefficients:

$6\sqrt{2} + 6\sqrt{2} - 5\sqrt{2}$

Step 4 — Combine like terms (all are "$\text{number} \times \sqrt{2}$"):

$(6 + 6 - 5)\sqrt{2} = 7\sqrt{2}$

Answer: $7\sqrt{2} \approx 9.899$.

Check: $3\sqrt{8} + 2\sqrt{18} - \sqrt{50} = 3(2.828) + 2(4.243) - 7.071 = 8.485 + 8.485 - 7.071 = 9.899$ ✓

The key rule

$a\sqrt{n} + b\sqrt{n} = (a + b)\sqrt{n}$ — you add the coefficients, the radical part stays the same. Just like $3x + 2x = 5x$.

You cannot add $\sqrt{2} + \sqrt{3}$ — different radicands don't combine.

Try it in the visualization

Adjust the coefficients. Each radical is simplified and shown in the same color. Like terms combine visually — bar lengths show the contribution of each term to the final answer.