Root Calculator

What is a root of a number?

The nth root of a number x is the value r that gives back x when it is multiplied by itself n times. The whole number n is the index or degree of the root and x is the radicand. Roots undo powers: since 3 × 3 × 3 × 3 × 3 = 243, the fifth root of 243 is 3.

r = x^(1/n) which means r^n = xThe radical sign √ with a small n written in its crook means the same thing.

The index decides how many real answers there are. When n is even and x is positive there are two real roots, one positive and one negative with the same size (both 2 and −2 square to 4). The positive one is called the principal root and it is the one this calculator returns. When n is odd there is exactly one real root, and it has the same sign as x.

Index 2 and index 3 are common enough to have their own tools, the square root calculator and the cube root calculator. This page covers every other index as well, including decimals.

How to use the root calculator

1. Enter the number you want the root of in Number.
2. Enter the degree of the root in Root: 2 for a square root, 3 for a cube root, 5 for a fifth root and so on.
3. Press Calculate to see the root.

Results are rounded to four decimal places. A whole-number answer is shown bare (the sixth root of 1,000,000 appears as 10), one that needs a single decimal is padded to two places (the square root of 2.25 appears as 1.50), and thousands are separated with commas.

To take the root of a fraction, enter the fraction as a decimal. The square root of 1/2 is found by typing 0.5 as the number and 2 as the root, which gives 0.7071. A fraction such as 1/4 typed into the Root box is not accepted; type 0.25 instead.

A fractional or negative root is allowed. A root of 0.5 squares the number (16 with root 0.5 gives 256), and a negative root gives a reciprocal (16 with root −2 gives 0.25). A root of 0 is not defined and is rejected.

Formulas and identities

nth root of x = x^(1/n)

nth root of (a × b) = (nth root of a) × (nth root of b)

nth root of (a ÷ b) = (nth root of a) ÷ (nth root of b)

nth root of (x^m) = x^(m/n)

The first line is how the calculator works: it raises the number to the power 1/n. The product and quotient rules are handy for simplifying by hand. For example, the square root of 144 ÷ 9 is 12 ÷ 3 = 4, and the fourth root of 16 × 81 is 2 × 3 = 6. The last rule links roots to fractional exponents, so the cube root of 5⁶ is 5^6/3 = 5² = 25.

Worked examples

A perfect fifth power. What is the fifth root of 243? Test small integers: 2⁵ = 32 is too small, 3⁵ = 243 is exact. Enter 243 and 5 to see 3.

An irrational root. The fourth root of 50 is 50^0.25 ≈ 2.6591. Check by multiplying it out: 2.6591⁴ ≈ 49.996, which rounds back to 50.

Growth rate. Money doubles in 10 years. The yearly growth factor g satisfies g¹⁰ = 2, so g is the tenth root of 2, about 1.0718. That is a steady gain of roughly 7.18% a year.

Music. In equal temperament each semitone multiplies the frequency by the twelfth root of 2, about 1.0595, so twelve semitones in a row give exactly one octave, a factor of 2.

Negative numbers and the nan result

Even roots of negative numbers are not real. No real number squared gives −16, so the square root of −16 lives in the complex numbers. For that reason, and because the calculator evaluates the power x^1/n directly, any negative input is reported as nan (not a number).

This also applies to odd roots, which do exist as real numbers. The fifth root of −32 is −2, because (−2)⁵ = −32. To get such a result, take the root of the positive value and put the sign back yourself: the fifth root of 32 is 2, so the fifth root of −32 is −2.

Checking your answer and common mistakes

• Swapped boxes. The first box is the number under the radical and the second is the index. The cube root of 8 is 2, but the eighth root of 3 is about 1.1472.
• Reading rounded output as exact. 1.4142 is the square root of 2 rounded to four places; 1.4142 × 1.4142 is 1.99996, not exactly 2.
• Confusing the root with a division. The cube root of 27 is 3, not 9. Dividing by 3 and taking the third root are different operations.
• Forgetting the check. Raise the answer to the power of the index. If you get the number you started with, the root is right.

For repeated multiplication in the other direction use the exponent calculator.