Practice Radical Operations in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick 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.

Showing a random 20 of 50 problems.

Example 1

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

Example 2

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

Example 3

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

Example 4

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

Example 5

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

Example 6

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

Example 7

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

Example 8

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

Example 9

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

Example 10

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

Example 11

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

Example 12

medium
Simplify 182\dfrac{\sqrt{18}}{\sqrt{2}}.

Example 13

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

Example 14

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

Example 15

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

Example 16

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

Example 17

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

Example 18

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

Example 19

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

Example 20

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