Simplify radical expressions by combining like terms

Expert intro: These are the kinds of problems where it helps to slow down for a second and sort the terms by their radical parts before doing anything else.

Detailed walkthrough

To simplify, first rewrite any radical that can be reduced, then combine only like radicals.

a

$2\sqrt{3} + 3\sqrt{7} + 3\sqrt{3}$

Combine the $\sqrt{3}$ terms:

$(2+3)\sqrt{3} + 3\sqrt{7} = 5\sqrt{3} + 3\sqrt{7}$

b

$\sqrt{11} + 3\sqrt{5} + \sqrt{11}$

$2\sqrt{11} + 3\sqrt{5}$

c

$-3\sqrt{2} + \sqrt{2} - 3\sqrt{5}$

$-2\sqrt{2} - 3\sqrt{5}$

d

$4\sqrt{8} + 2\sqrt{7} - \sqrt{8} - 4\sqrt{7}$

First simplify $\sqrt{8} = 2\sqrt{2}$:

$4\sqrt{8} - \sqrt{8} = 3\sqrt{8} = 6\sqrt{2}$

Now combine:

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

e

$4\sqrt{5} - \sqrt{2} + 4\sqrt{5} - 3\sqrt{2}$

$8\sqrt{5} - 4\sqrt{2}$

f

$9\sqrt{3} + 3\sqrt{8} - \sqrt{3} + \sqrt{8}$

Simplify $\sqrt{8} = 2\sqrt{2}$:

$9\sqrt{3} - \sqrt{3} + 3\sqrt{8} + \sqrt{8} = 8\sqrt{3} + 4\sqrt{8}$

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

So the final answer is

$8\sqrt{3} + 8\sqrt{2}$

Answers

• a) $5\sqrt{3} + 3\sqrt{7}$
• b) $2\sqrt{11} + 3\sqrt{5}$
• c) $-2\sqrt{2} - 3\sqrt{5}$
• d) $6\sqrt{2} - 2\sqrt{7}$
• e) $8\sqrt{5} - 4\sqrt{2}$
• f) $8\sqrt{3} + 8\sqrt{2}$

💡 Pitfall guide

Two easy slips show up here: forgetting to simplify radicals like $\sqrt{8}$ before combining, and trying to combine unlike terms such as $\sqrt{3}$ and $\sqrt{7}$. Keep the radical part the same before you add or subtract coefficients.

🔄 Real-world variant

If a coefficient is negative, carry the sign with the term and combine as usual. If a radical can be simplified in more than one step, reduce it fully before grouping terms. That often reveals extra like terms you would miss at first glance.

🔍 Related terms

combine like terms, simplest radical form, square root simplification