Radical Expressions Worksheets

Simplify square roots, combine like radicals and multiply radical expressions. Grades 9–11.

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What are radical expressions?

A radical expression contains a root, most often a square root such as 50\sqrt{50}. A square root is in simplest form when the number under the root (the radicand) has no perfect-square factor other than 1. Simplifying relies on the product rule ab=a⋅b\sqrt{ab} = \sqrt{a} \cdot \sqrt{b} for non-negative numbers: split off the largest perfect square and take its root.

Radicals with the same radicand are like radicals and combine the same way like terms do: 32+62=923\sqrt{2} + 6\sqrt{2} = 9\sqrt{2}, just as 3x+6x=9x3x + 6x = 9x.

  • Easy — simplify a single square root, such as 75\sqrt{75}.
  • Medium — simplify, then add or subtract like radicals. The answer can be negative.
  • Hard — multiply two radical terms, then simplify the result.

Worked examples

Easy: 75\sqrt{75}

The largest perfect square that divides 75 is 25.

75=25⋅3=25⋅3=53\sqrt{75} = \sqrt{25 \cdot 3} = \sqrt{25}\cdot\sqrt{3} = 5\sqrt{3}

Medium: 32+2183\sqrt{2} + 2\sqrt{18}

The two radicals are not alike yet, so simplify 18=9⋅2=32\sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2} first.

32+218=32+2⋅32=32+62=923\sqrt{2} + 2\sqrt{18} = 3\sqrt{2} + 2 \cdot 3\sqrt{2} = 3\sqrt{2} + 6\sqrt{2} = 9\sqrt{2}

Hard: 36×233\sqrt{6} \times 2\sqrt{3}

Multiply the numbers outside the roots together and the radicands together, then simplify.

36×23=618=6⋅32=1823\sqrt{6} \times 2\sqrt{3} = 6\sqrt{18} = 6 \cdot 3\sqrt{2} = 18\sqrt{2}

Check your answers

A calculator makes a quick check: 75≈8.660\sqrt{75} \approx 8.660 and 53≈8.6605\sqrt{3} \approx 8.660. Matching decimals mean the simplification is right; the simplified radical is still the exact answer.

Common mistakes

  • Adding radicands. 2+8\sqrt{2} + \sqrt{8} is not 10\sqrt{10}. Simplify first: 8=22\sqrt{8} = 2\sqrt{2}, so the sum is 323\sqrt{2}.
  • Not using the largest perfect square. 72=218\sqrt{72} = 2\sqrt{18} is true but not finished, because 18 still contains 9. The simplest form is 626\sqrt{2}.
  • Mixing up what multiplies. In 36×233\sqrt{6} \times 2\sqrt{3}, the 3 and 2 multiply to 6 and the radicands multiply to 18. The coefficients are not added.
  • Dropping a negative result. 43−327=43−93=−534\sqrt{3} - 3\sqrt{27} = 4\sqrt{3} - 9\sqrt{3} = -5\sqrt{3}. A negative coefficient is a perfectly good answer.

Sample problems

These problems come from the same generator as the worksheet above, three at each difficulty level. Press Generate for a fresh set.

Easy

  1. 54=\sqrt{54} =
  2. 80=\sqrt{80} =
  3. 32=\sqrt{32} =

Medium

  1. 27+328=2\sqrt{7} + 3\sqrt{28} =
  2. 5−320=\sqrt{5} - 3\sqrt{20} =
  3. 52+432=5\sqrt{2} + 4\sqrt{32} =

Hard

  1. 22×37=2\sqrt{2} \times 3\sqrt{7} =
  2. 42×35=4\sqrt{2} \times 3\sqrt{5} =
  3. 43×37=4\sqrt{3} \times 3\sqrt{7} =
Show answers
  1. 3√6
  2. 4√5
  3. 4√2
  4. 8√7
  5. -5√5
  6. 21√2
  7. 6√14
  8. 12√10
  9. 12√21

Tips for teachers and parents

  • Keep a list of perfect squares (4, 9, 16, 25, 36, 49, 64, 81, 100) visible while students work on Easy; after a few worksheets they will not need it.
  • On Medium, ask students to rewrite every radical in simplest form on its own line before combining anything.
  • Decimal checks with a calculator are a good self-marking tool for homework.

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