Formula
ⁿ√x = x^(1/n)√(a·b) = √a · √b (used to simplify radicals) How it works
The square root of x is the number that multiplies by itself to give x: √49 = 7 because 7 × 7 = 49. A cube root works with three copies (∛27 = 3) and an nth root with n copies.
Most square roots are irrational — their decimals never end or repeat — so teachers often want the simplified radical form instead. To simplify √72, find the largest perfect square that divides 72, which is 36. Then √72 = √36 × √2 = 6√2.
Odd roots of negative numbers are fine (∛−8 = −2), but even roots of negative numbers have no real answer, because any real number raised to an even power is non-negative. Those roots live in the complex numbers, written with i = √−1.
Frequently asked questions
Does a square root have two answers?
The equation x² = 49 has two solutions, 7 and −7. The symbol √49 means the principal (non-negative) root only, which is 7.
How can I estimate a square root by hand?
Find the perfect squares on either side. √50 is between √49 = 7 and √64 = 8, and much closer to 7, so it’s a bit over 7 (actually 7.071).