Root Calculator

A root undoes a power. The square root of x is the number that, multiplied by itself, gives x, so √49 = 7 because 7 × 7 = 49. The cube root of x is the number that gives x when used three times as a factor, so ∛125 = 5. In general, the nth root of x, written ⁿ√x, is the number r for which rⁿ = x. Roots can also be written as fractional exponents: ⁿ√x = x^(1/n), which is why √x = x^0.5.

Every positive number has two square roots, one positive and one negative, since (−7)² is also 49. The √ symbol always means the positive one, called the principal root. Negative numbers have no real square root, because any real number squared is zero or positive; their square roots are imaginary, such as √−9 = 3i. Odd roots behave differently: ∛−8 = −2, because (−2)³ = −8.

Numbers like 16, 81 and 1,000 are perfect powers with whole-number roots. Most roots, such as √2, are irrational: their decimals never end or repeat. For those, the exact value is often written in simplified radical form by taking perfect powers out from under the sign. √72 = √(36 × 2) = 6√2, and ∛54 = ∛(27 × 2) = 3∛2.

Estimating a Root

Before calculators, roots were found by educated guessing and refinement, and the same idea still makes a good mental check. Start by locating the number between two perfect squares. 27 sits between 25 and 36, so √27 is between 5 and 6, and closer to 5.

A fast refinement is the Babylonian method, also called Heron's method. Divide the number by your guess, then average the guess and the quotient. If the guess is too small, the quotient is too big, and their average lands closer to the true root. Repeat until the digits stop changing. The table shows √27 converging from a first guess of 5.

Estimating an nth Root

The Babylonian method is a special case of Newton's method, which works for any index. To estimate ⁿ√x, start from a guess g, then replace it with ((n − 1)·g + x ÷ gⁿ⁻¹) ÷ n. The new value is a weighted average that gives the old guess a weight of n − 1 and the quotient x ÷ gⁿ⁻¹ a weight of 1. Once the guess is close, each step roughly doubles the number of correct digits.

For example, to estimate the fifth root of 100, notice that 2⁵ = 32 and 3⁵ = 243, so the answer is between 2 and 3. Starting from 2.5, one step gives 2.512, the next gives 2.5118864, and the third agrees with the true value 2.5118864315 to ten digits.

Logarithms offer another route: ⁿ√x = 10^(log₁₀ x ÷ n). Since log₁₀ 100 = 2, the fifth root of 100 is 10^0.4 ≈ 2.5119. This was the standard technique when people used printed log tables or slide rules. This calculator instead uses exact integer arithmetic to find each digit, so the 20 digits it shows are all correct.

ⁿ√x = x^(1/n), xₖ₊₁ = xₖ − (xₖⁿ − x) ÷ (n·xₖⁿ⁻¹)

n

the index of the root (2 for square, 3 for cube)

xₖ

the current estimate; each Newton step usually doubles the correct digits

Frequently asked questions