Square Roots & Radicals · Grades 7–11

Square Roots & Radicals Worksheets

Perfect squares first, then simplifying radicals like √72 = 6√2 by pulling out the biggest square factor. Printable square root worksheets.

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What it is

A square root of a number is the value that, squared, gives it. To simplify a radical, pull out the largest perfect-square factor.

Why it matters

Square roots appear in the Pythagorean theorem, the quadratic formula, and geometry. Exact square roots of perfect squares are a core Grade 7 skill; simplifying a radical to a√b builds on that in Grade 11.

How to do it

  1. Find the largest perfect square that divides the number.
  2. Write the number as (perfect square) × (the rest).
  3. Take the square root of the perfect square outside the radical.

Examples by level

  • Beginner18\sqrt{18} =323\sqrt{2}
  • Intermediate18\sqrt{18} =323\sqrt{2}
  • Advanced48\sqrt{48} =434\sqrt{3}

Examples are generated by the same engine as the worksheets, so they're always mathematically correct. Built to avoid repeats, not reshuffle the same handful of questions.

Common mistakes

  • Not pulling out the largest perfect-square factor (leaving √72 as 2√18).
  • Adding the outside and inside numbers.
  • Thinking √(a + b) equals √a + √b.

Tips

  • Learn the perfect squares up to 225.
  • Factor out the LARGEST square, or you'll have to simplify again.

For parents

To simplify, find the biggest perfect square that divides the number, then take its square root outside the radical.

For teachers

Have students memorize the perfect squares to 15²; it makes spotting the square factor almost automatic.

Key vocabulary

radicand
the number under the radical sign
perfect square
the square of a whole number, like 1, 4, 9, 16

Frequently asked questions

How do you simplify a square root?
Factor out the largest perfect square, take its root outside the radical, and leave the rest inside, e.g., √72 = √(36·2) = 6√2.
What is a perfect square?
A number that is the square of a whole number, like 1, 4, 9, 16, 25, its square root is exact.

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