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How to Put Nth Root in Calculator

Reviewed by Calculator Editorial Team

Calculating nth roots is a fundamental mathematical operation that appears in many scientific and engineering applications. This guide explains how to perform nth root calculations using different calculator methods, provides the mathematical formula, offers worked examples, and addresses common mistakes.

How to Calculate Nth Root

The nth root of a number x is a value that, when raised to the power of n, gives x. There are several methods to calculate nth roots using calculators:

  1. Use the calculator's built-in root function (often labeled as "y√x" or "n√x")
  2. Use the exponentiation function with fractional exponents
  3. Use logarithms to solve for roots

Most scientific and graphing calculators have a dedicated root function that allows you to specify both the radicand (the number under the root) and the index (the number indicating the root).

Calculator Methods

Method 1: Using the Root Function

Most modern calculators have a dedicated root function. Here's how to use it:

  1. Enter the radicand (the number under the root)
  2. Press the root function button (often labeled as "y√x" or "n√x")
  3. Enter the index (the number indicating the root)
  4. Press the equals button to get the result

Method 2: Using Exponentiation

You can also calculate roots using exponentiation with fractional exponents:

  1. Enter the radicand
  2. Press the exponentiation button (often labeled as "^" or "x^y")
  3. Enter the reciprocal of the index (1 divided by the root number)
  4. Press the equals button to get the result

Method 3: Using Logarithms

For more complex calculations, you can use logarithms to solve for roots:

  1. Take the natural logarithm of the radicand
  2. Divide the result by the index
  3. Exponentiate the result to get the nth root

The Formula

The general formula for calculating the nth root of a number x is:

n√x = x^(1/n)

Where:

  • n is the index (the number indicating the root)
  • x is the radicand (the number under the root)

For example, the cube root of 27 is 27^(1/3) = 3.

Worked Examples

Example 1: Square Root

Calculate the square root of 64 using the root function:

  1. Enter 64
  2. Press the root function (√x)
  3. Enter 2 (for square root)
  4. Result: 8

Example 2: Cube Root

Calculate the cube root of 125 using exponentiation:

  1. Enter 125
  2. Press the exponentiation button (^)
  3. Enter 1/3 (reciprocal of 3)
  4. Result: 5

Example 3: Fifth Root

Calculate the fifth root of 32 using logarithms:

  1. Calculate ln(32) ≈ 3.4657
  2. Divide by 5: 3.4657/5 ≈ 0.6931
  3. Exponentiate: e^0.6931 ≈ 2

Common Mistakes

When calculating nth roots, be aware of these common errors:

  • Confusing the index and radicand positions
  • Using the wrong root function (square root instead of cube root)
  • Forgetting to enter the index for the root function
  • Rounding errors in intermediate steps when using logarithms

Always double-check your inputs and verify the result by raising it to the power of n to ensure it equals the original radicand.

FAQ

What is the difference between square root and nth root?
A square root is a special case of nth root where n=2. The square root of a number x is x^(1/2). Other nth roots include cube roots (n=3), fourth roots (n=4), and so on.
Can I calculate roots of negative numbers?
Yes, but the results depend on the index. Even-indexed roots (like square roots) of negative numbers are not real numbers, while odd-indexed roots (like cube roots) can be negative.
How accurate are calculator root calculations?
Most scientific calculators provide accurate results to at least 10 decimal places. For more precise calculations, consider using programming languages or specialized mathematical software.
What if my calculator doesn't have a root function?
You can still calculate roots using exponentiation with fractional exponents or logarithms, as described in the methods section.