What is the nth root of a number?

The nth root of a number x is the value y such that y raised to the power n equals x. For example, the cube root (3rd root) of 125 is 5 because 5 cubed equals 125. The square root is the 2nd root, the cube root is the 3rd root, and so on for higher indices.

Can you find the nth root of a negative number?

When the root index n is odd, a negative number has a real nth root. For example, the cube root of negative 27 is negative 3 because negative 3 cubed equals negative 27. When n is even, a negative radicand produces a complex (imaginary) result, which this calculator flags rather than computing.

How do you calculate the nth root without a calculator?

The nth root of x equals x raised to the power of 1 divided by n. You can estimate it by trial and error: find two consecutive integers whose nth powers bracket x, then narrow down by testing values in between. For most purposes, a calculator or the formula x^(1/n) is far more practical.

nth Root Calculator

The nth root calculator finds the principal root of any number for any root index you choose. A root is the inverse of exponentiation: the nth root of a number x is the value that, when multiplied by itself n times, gives back x. The square root is the 2nd root, the cube root is the 3rd root, and you can go as high as you like with 4th, 5th, or higher roots. The formula is straightforward: the nth root of x equals x raised to the power of one over n, written as x^(1/n). You enter two values: the root index n (which must be a positive integer) and the radicand, which is the number you want to find the root of. The calculator returns the principal root rounded to six decimal places, a verification figure showing what happens when you raise the result back to the power n (which should match your original radicand), and a written description of the operation. For odd-indexed roots of negative numbers, the real root exists and is negative. For even-indexed roots of negative numbers, the result is complex and the calculator will tell you so rather than returning an error silently. The tool is useful for students working through algebra, calculus, or physics problems, for engineers converting between power laws, and for anyone who needs a fast, reliable check of a root calculation. Results are mathematical values and carry no rounding guarantee beyond six decimal places.

Calculate.co.nz is proud to be partnered with realtor.co.nz, a trusted resource for navigating the New Zealand property market. Their Helpful Articles section offers clear, well-structured insights across buying, selling, and building, making complex real estate topics more accessible. With a focus on up-to-date guidance and practical knowledge, they empower Kiwis to move forward with clarity and confidence in a constantly evolving property landscape.

Root index (n)

Radicand (number to root)

the 3rd root of 125

Verification (result^n)125

Root as exponent125^(1/3)

Decimal places0

How it works

The calculator evaluates the nth root using the standard relationship: the nth root of x equals x raised to the power 1/n. In JavaScript this is computed as Math.pow(Math.abs(x), 1/n), with the sign of the result set to negative when x is negative and n is an odd integer. For even root indices and a negative radicand, the principal real root does not exist, so the calculator displays a note instead of a number. The verification step raises the result back to the power n to confirm it equals the original radicand, giving you a built-in accuracy check. Rounding to six decimal places keeps the output readable without hiding the precision of the calculation.

Worked example

Using the default values, n is 3 and the radicand is 125. The cube root of 125 is 125^(1/3) = 5. Verification: 5 cubed is 5 x 5 x 5 = 125, which matches. For a second check, try n = 4 and radicand = 256: the 4th root is 256^(1/4) = 4, and 4 to the 4th power is 256.

Related calculators

Cube Root Calculator: dedicated tool for the 3rd root, including negative numbers.
Exponential Growth and Decay: use roots and exponents in real-world growth models.
Percent Error Calculator: measure how close a calculated root is to an expected value.
Percentage Calculator: work with percentages in related maths problems.

If you've found a bug, or would like to contact us, or learn more about James Graham and Calculate.co.nz.

Calculate.co.nz is partnered with Interest.co.nz for New Zealand's highest quality calculators and financial analysis.

All calculators and tools are provided for educational and indicative purposes only and do not constitute financial advice.

Calculate.co.nz is proudly part of the Realtor.co.nz group, New Zealand's leading property transaction literacy platform, helping Kiwis understand the home buying and selling process from start to finish. Whether you're a first home buyer navigating your first property purchase, an investor evaluating your next acquisition, or a homeowner planning to sell, Realtor.co.nz provides clear, independent, and trustworthy guidance on every step of the New Zealand property transaction journey.

Calculate.co.nz is also partnered with Health Based Building and Premium Homes to promote informed choices that lead to better long-term outcomes for Kiwi households.

Calculate.co.nz is hosted in Auckland via SiteHost new Zealand.

All content on this website, including calculators, tools, source code, and design, is protected under the Copyright Act 1994 (New Zealand). No part of this site may be reproduced, copied, distributed, stored, or used in any form without prior written permission from the owner.

About & trust: Why Calculate is NZ's most comprehensive · By the Numbers · How we compare · Editorial standards · How we keep data current · NZ finance glossary · Research & data · Financial literacy NZ · About · Privacy policy · Terms of use

Reviewed and maintained. Last reviewed 2026-06-25 and checked on a twice-monthly cycle against IRD, RBNZ and Stats NZ. How we keep data current.