Radical Operations Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Radical Operations.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Adding, subtracting, and multiplying expressions that contain radicals. Like terms (same radicand) can be combined for addition and subtraction; for multiplication, use a⋅b=ab\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}.

Treat simplified radicals like variables: 35+25=553\sqrt{5} + 2\sqrt{5} = 5\sqrt{5} works just like 3x+2x=5x3x + 2x = 5x. You can only combine radicals with the SAME radicand. Multiplication is more flexible since a⋅b=ab\sqrt{a} \cdot \sqrt{b} = \sqrt{ab} always works.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Combine only matching radicands for +/−+/-, but multiply radicands freely under one root.

Common stuck point: The procedure for radical operations is the easy part; the trap is adding the radicands when multiplying. Asking "For ++ or −-, do the radicands match — and have I simplified first to check?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: For ++ or −-, do the radicands match — and have I simplified first to check?

Worked Examples

Example 1

easy
Simplify 35+753\sqrt{5} + 7\sqrt{5}.

Answer

10510\sqrt{5}

First step

1
Step 1: Both terms have the same radicand 5\sqrt{5}, so they are like radicals.

Full solution

  1. 2
    Step 2: Add coefficients: (3+7)5=105(3 + 7)\sqrt{5} = 10\sqrt{5}.
  2. 3
    Check: Think of 5\sqrt{5} as a variable: 3a+7a=10a3a + 7a = 10a ✓
Like radicals (same radicand and same index) can be combined by adding or subtracting their coefficients, just like combining like terms in algebra.

Example 2

medium
Simplify 12+27\sqrt{12} + \sqrt{27}.

Example 3

medium
Expand 3(12+3)\sqrt{3}(\sqrt{12}+\sqrt{3}).

Example 4

medium
Rationalize the denominator: 15\dfrac{1}{\sqrt{5}}.

Example 5

hard
Expand and simplify (8+2)(18−2)(\sqrt{8}+\sqrt{2})(\sqrt{18}-\sqrt{2}).

Example 6

hard
Rationalize and simplify 55+3\dfrac{\sqrt{5}}{\sqrt{5}+\sqrt{3}}.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium
Multiply 6⋅10\sqrt{6} \cdot \sqrt{10} and simplify.

Example 2

hard
Expand and simplify (3+2)(3−2)(\sqrt{3} + \sqrt{2})(\sqrt{3} - \sqrt{2}).

Example 3

easy
Simplify 35+253\sqrt{5} + 2\sqrt{5}.

Example 4

easy
Simplify 73−437\sqrt{3} - 4\sqrt{3}.

Example 5

easy
Multiply 3⋅12\sqrt{3}\cdot\sqrt{12}.

Example 6

easy
Multiply 2⋅8\sqrt{2}\cdot\sqrt{8}.

Example 7

easy
Simplify 6⋅6\sqrt{6}\cdot\sqrt{6}.

Example 8

easy
Simplify 52+25\sqrt{2} + \sqrt{2}.

Example 9

easy
Multiply 5⋅7\sqrt{5}\cdot\sqrt{7}.

Example 10

easy
Simplify 27+37−72\sqrt{7} + 3\sqrt{7} - \sqrt{7}.

Example 11

medium
Simplify 12+27\sqrt{12} + \sqrt{27}.

Example 12

medium
Expand 2(6+2)\sqrt{2}(\sqrt{6} + \sqrt{2}).

Example 13

medium
Expand (3+2)(3−2)(\sqrt{3} + 2)(\sqrt{3} - 2).

Example 14

medium
Expand (5+2)2(\sqrt{5} + \sqrt{2})^2.

Example 15

medium
Simplify 8+18−2\sqrt{8} + \sqrt{18} - \sqrt{2}.

Example 16

medium
Multiply 26⋅322\sqrt{6}\cdot3\sqrt{2}.

Example 17

medium
Expand (23−1)(3+4)(2\sqrt{3} - 1)(\sqrt{3} + 4).

Example 18

medium
Simplify 20+35\sqrt{20} + 3\sqrt{5}.

Example 19

medium
Multiply 10⋅15\sqrt{10}\cdot\sqrt{15} and simplify.

Example 20

challenge
Expand (6−2)2(\sqrt{6} - \sqrt{2})^2 and simplify.

Example 21

challenge
Simplify 50+32−8\sqrt{50} + \sqrt{32} - \sqrt{8}.

Example 22

challenge
Expand and simplify (x+3)(x−3)(\sqrt{x} + 3)(\sqrt{x} - 3) for x≥0x\ge0.

Example 23

easy
Simplify 411+6114\sqrt{11} + 6\sqrt{11}.

Example 24

easy
Simplify 96−269\sqrt{6} - 2\sqrt{6}.

Example 25

easy
Multiply 11⋅11\sqrt{11}\cdot\sqrt{11}.

Example 26

easy
Simplify 2+2+2\sqrt{2}+\sqrt{2}+\sqrt{2}.

Example 27

easy
Multiply 2⋅18\sqrt{2}\cdot\sqrt{18}.

Example 28

easy
Simplify 610−6106\sqrt{10} - 6\sqrt{10}.

Example 29

medium
Simplify 45+20\sqrt{45}+\sqrt{20}.

Example 30

medium
Simplify 72−32\sqrt{72}-\sqrt{32}.

Example 31

medium
Expand (7+1)(7−1)(\sqrt{7}+1)(\sqrt{7}-1).

Example 32

medium
Expand (7+2)2(\sqrt{7}+2)^2.

Example 33

medium
Multiply 46⋅2154\sqrt{6}\cdot 2\sqrt{15} and simplify.

Example 34

medium
Simplify 502\dfrac{\sqrt{50}}{\sqrt{2}}.

Example 35

medium
Expand (25+3)(5−3)(2\sqrt{5}+\sqrt{3})(\sqrt{5}-\sqrt{3}).

Example 36

medium
Simplify 75−12+48\sqrt{75}-\sqrt{12}+\sqrt{48}.

Example 37

hard
Rationalize and simplify 62−3\dfrac{6}{2-\sqrt{3}}.

Example 38

hard
Simplify (32−8)2(3\sqrt{2}-\sqrt{8})^2.

Example 39

hard
Simplify 200+98−8\sqrt{200}+\sqrt{98}-\sqrt{8}.

Example 40

hard
Solve for aa: a2+32=102a\sqrt{2}+3\sqrt{2}=10\sqrt{2}.

Example 41

challenge
Simplify (2+3+6)2−(2+3)2(\sqrt{2}+\sqrt{3}+\sqrt{6})^2-(\sqrt{2}+\sqrt{3})^2.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

simplifying radicalsexpressions