Module 8 · Radicals · Lesson 33 of 39
Square roots and simplifying radicals
Roots undo powers. Pull perfect squares out of the root.
In this lesson · about 14 minutes
- Find square and cube roots
- Simplify a square root by factoring out a perfect square
A square root undoes squaring: √9 = 3 because 3 · 3 = 9. The number under the root sign is the radicand. For any positive number there's exactly one positive square root, and that's what √ means.
A cube root asks which number used 3 times gives the radicand: ∛8 = 2 and ∛(−8) = −2. Roots can also be written as fractional exponents: (−8)1/3 = ∛(−8).
Simplifying
If the radicand isn't a perfect square, look for a perfect-square factor and take its root out: √40 = √4 · 10 = √4 · √10 = 2√10.
Try it · interactive
Square root explorer
Slide the radicand. Perfect squares make a full square grid; the others simplify (or don't).
√72 = 6√2
≈ 8.4853
72 is between the perfect squares 64 and 81, so √72 is between 8 and 9. The biggest perfect square that divides 72 is 36: √72 = √36 · √2 = 6√2.
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