Square Root Calculator — √ and Nth Root

Square root: √x = x^(1/2)
Cube root: ³√x = x^(1/3)
Nth root: x^(1/n)

Perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144
Perfect cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000

Every positive number has two square roots: +√x and -√x. The √ symbol returns only the principal (positive) root by convention. Roots are fractional exponents: √x = x^(1/2).

Square root

=SQRT(A1)

Cube root

=A1^(1/3)

Nth root (n in B1)

=A1^(1/B1)

Square Roots and Nth Roots ExplainedSearch YouTube tutorials →

Roots as Fractional Exponents

One of the most useful insights in algebra: roots are just fractional exponents. The square root of x is x^(1/2). The cube root is x^(1/3). The fourth root is x^(1/4). This connection is more than notation — it means roots follow all the same algebraic rules as exponents.

For example: √(x^4) = (x^4)^(1/2) = x^(4×1/2) = x^2. Or: (√8)^3 = 8^(1/2×3) = 8^(3/2) = (8^3)^(1/2) = √512 ≈ 22.6. Understanding this unification of roots and exponents makes simplifying algebraic expressions much faster.

Fractional exponents also explain negative roots naturally: x^(-1/2) = 1/√x. A negative fractional exponent is a reciprocal root. This appears in physics formulas, statistics (standard error ∝ n^(-1/2)), and engineering regularly.

Estimating Square Roots Mentally

Find the two perfect squares that bracket your number, then interpolate. Example: estimate √50. √49 = 7 and √64 = 8. Since 50 is very close to 49, the root is just above 7 — specifically 7.07. For more precision: (50-49)/(64-49) = 1/15 ≈ 0.07 above 7, giving 7.07. Actual: 7.071.

Perfect squares worth memorizing: 1²=1, 2²=4, 3²=9, 4²=16, 5²=25, 6²=36, 7²=49, 8²=64, 9²=81, 10²=100, 11²=121, 12²=144, 13²=169, 14²=196, 15²=225. Knowing these allows instant recognition of perfect squares and rapid estimation of non-perfect square roots.

Frequently Asked Questions

If x is not a perfect square, √x is irrational — its decimal expansion never ends or repeats. √2 ≈ 1.41421356... cannot be expressed as a fraction of two integers. This was proven by ancient Greek mathematicians and reportedly caused a philosophical crisis for Pythagoreans who believed all numbers were rational.

Not in real numbers. In complex numbers, √(-1) is defined as the imaginary unit i. So √(-4) = 2i. Complex numbers have extensive applications in electrical engineering, physics, and signal processing, but are beyond the scope of this calculator which handles real numbers only.

√4 = 2 (principal positive root, by convention). ±√4 = ±2 (both positive and negative roots). In the quadratic formula, ± is used because both roots of a quadratic equation are valid solutions. When solving x² = 4, the answer is x = ±2. But √4 as a function returns only +2.

Factor out perfect squares. √72 = √(36×2) = 6√2. √48 = √(16×3) = 4√3. The rule √(a×b) = √a × √b applies to non-negative numbers. A square root is fully simplified when no perfect square factor remains under the radical sign.