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How to Take The Tenth Root on A Calculator

Reviewed by Calculator Editorial Team

The tenth root of a number is a value that, when raised to the power of 10, gives the original number. This is a specialized calculation that appears in advanced mathematics, engineering, and scientific research. This guide explains how to perform this calculation accurately using standard calculators.

What is the Tenth Root?

The tenth root of a number \( x \) is a number \( y \) such that \( y^{10} = x \). In mathematical terms, this is expressed as:

\( y = \sqrt[10]{x} \)

This calculation is less common than square roots or cube roots, but it appears in specialized fields like:

  • Advanced mathematical analysis
  • Scientific simulations
  • Engineering calculations
  • Financial modeling

Most standard calculators don't have a direct tenth root button, so you'll need to use exponentiation functions.

Methods to Calculate the Tenth Root

There are several approaches to calculate the tenth root using a standard calculator:

  1. Using the exponentiation function (most common)
  2. Using logarithms (for more complex calculations)
  3. Using repeated multiplication (for educational purposes)

The exponentiation method is the most straightforward for most users.

Step-by-Step Guide

Using the Exponentiation Function

  1. Enter the number you want to find the tenth root of
  2. Press the "y^x" or "exp" button (varies by calculator model)
  3. Enter "0.1" (which is 1/10)
  4. Press the equals (=) button

Note: Some calculators may require you to enter "1/10" instead of "0.1".

Using Logarithms

  1. Take the natural logarithm (ln) of the number
  2. Divide the result by 10
  3. Take the exponential (e^x) of the result

This method is more complex but useful when working with very large numbers.

Worked Examples

Example 1: Finding the Tenth Root of 1000

Using the exponentiation method:

  1. Enter 1000
  2. Press y^x
  3. Enter 0.1
  4. Press =

The result is approximately 1.77827941.

Result

≈ 1.77827941

Verification: \( 1.77827941^{10} ≈ 1000 \)

Example 2: Finding the Tenth Root of 0.001

Using the exponentiation method:

  1. Enter 0.001
  2. Press y^x
  3. Enter 0.1
  4. Press =

The result is approximately 0.562341325.

Result

≈ 0.562341325

Verification: \( 0.562341325^{10} ≈ 0.001 \)

FAQ

Why don't standard calculators have a tenth root button?

The tenth root is less commonly used than square roots or cube roots, so it's not included on most basic calculators. You can still calculate it using exponentiation functions.

What's the difference between the tenth root and the square root?

The tenth root finds a number that, when raised to the 10th power, gives the original number. The square root finds a number that, when squared, gives the original number. The tenth root is a more specialized calculation.

Can I calculate the tenth root of negative numbers?

No, the tenth root of negative numbers is not a real number. It's only defined for positive real numbers and complex numbers.

How accurate are calculator results for tenth roots?

Most scientific calculators provide results accurate to at least 10 decimal places. For more precise calculations, you might need specialized software.