Root Calculator

What Is a Root?

A root is the reverse of an exponent. The nth root of a number a is the value b that, multiplied by itself n times, gives back a: if bⁿ = a, then b = ⁿ√a. The two most common cases have their own names — the square root (n = 2) and the cube root (n = 3) — and this root calculator handles those and any higher root in one place. For example, √49 = 7 because 7 × 7 = 49, and ³√64 = 4 because 4 × 4 × 4 = 64.

Roots as Fractional Exponents

Every root can be written as a power: ⁿ√a = a^(1/n). That is how calculators evaluate them internally, and it lets you use all the usual exponent rules — see the Exponent Calculator. It also explains why even roots of negative numbers are not real numbers: no real number multiplied by itself gives a negative result. Odd roots of negatives are fine, so ³√−8 = −2.

Estimating a Square Root by Hand

Before calculators, people used a repeat-and-refine trick sometimes called the Babylonian method:

1. Guess a number b close to the answer.
2. Divide your number a by b to get c.
3. Average b and c to get a better guess, and repeat.

To find √27, guess 5, then 27 ÷ 5 = 5.4, average to 5.2, then 27 ÷ 5.2 ≈ 5.192, average to about 5.196. In a few rounds you have three correct decimals. The same idea extends to higher roots with a small modification: the quotient becomes a ÷ bⁿ⁻¹ and the average weights b by (n − 1).

Perfect Squares and Perfect Cubes

Numbers whose square root is a whole number are called perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 and so on. Perfect cubes follow the same pattern with 1, 8, 27, 64, 125. Memorizing the first few makes it much faster to spot simplifications — √72, for instance, equals 6√2 because 72 = 36 × 2.

Where Roots Show Up

Roots appear in the Pythagorean theorem for distances (try the Pythagorean Theorem Calculator), in the quadratic formula, in standard deviation, and in geometry whenever an area or volume must be turned back into a length.