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How to Input Nth Root in Calculator

Reviewed by Calculator Editorial Team

Calculating nth roots is a fundamental mathematical operation that appears in many fields including algebra, physics, and engineering. This guide explains how to properly input nth root calculations in various types of calculators, including scientific, programming, online, and mobile calculators.

Basic Methods to Input Nth Root

The most straightforward way to calculate an nth root is to use the exponentiation function available on most calculators. The nth root of a number x can be calculated as x raised to the power of 1/n.

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

For example, to calculate the cube root of 27, you would input 27^(1/3).

Example Calculation

Find the 5th root of 32.

Input: 32^(1/5)

Result: 2 (since 2^5 = 32)

This method works on all types of calculators that have an exponentiation function, including basic, scientific, and programming calculators.

Using Scientific Calculators

Scientific calculators typically have a dedicated key for exponentiation, often labeled as "x^y" or "^". Here's how to use it:

  1. Enter the number you want to find the root of (the radicand)
  2. Press the exponentiation key (x^y or ^)
  3. Enter 1 divided by the root number (1/n)
  4. Press the equals (=) key to get the result

For example, to find the square root of 16 on a scientific calculator:

  1. Enter 16
  2. Press x^y
  3. Enter 1/2 (which is 0.5)
  4. Press = to get 4

Some scientific calculators also have a dedicated square root key (√) and cube root key (∛), but these only handle specific roots. For other roots, you'll need to use the exponentiation method.

Programming Calculators

Programming calculators, often used by engineers and programmers, typically have more advanced functions for handling roots. These calculators often have a "y√x" function that directly calculates the nth root.

  1. Enter the root number (n)
  2. Press the y√x function key
  3. Enter the radicand (x)
  4. Press the equals (=) key to get the result

For example, to find the 4th root of 16 on a programming calculator:

  1. Enter 4
  2. Press y√x
  3. Enter 16
  4. Press = to get 2

If your programming calculator doesn't have a y√x function, you can still use the exponentiation method described in the basic methods section.

Online Calculators

Online calculators provide a convenient way to calculate nth roots without needing physical hardware. Most online calculators have an exponentiation function that can be used to find roots.

  1. Enter the radicand (x)
  2. Type the caret symbol (^) or use the exponentiation button
  3. Enter (1/n)
  4. Press the equals button or hit enter to get the result

For example, to find the 6th root of 64 using an online calculator:

  1. Enter 64
  2. Type ^
  3. Enter (1/6)
  4. Press = to get 2

Some online calculators have a dedicated root function that you can use by entering the root number first, then the radicand.

Mobile Apps

Many mobile calculator apps follow the same principles as desktop calculators. The process is generally the same:

  1. Open the calculator app
  2. Enter the radicand
  3. Use the exponentiation function (often labeled as x^y or ^)
  4. Enter 1 divided by the root number
  5. Tap equals to get the result

For example, to find the cube root of 216 using a mobile calculator:

  1. Open the calculator
  2. Enter 216
  3. Press x^y
  4. Enter 1/3
  5. Tap = to get 6

Some mobile calculator apps have a dedicated root function that you can use by entering the root number first, then the radicand.

Common Mistakes to Avoid

When calculating nth roots, there are several common mistakes that users make:

  • Incorrect order of operations: Remember that exponentiation has higher precedence than division, so you need to use parentheses when calculating 1/n.
  • Using the wrong root: Make sure you're using the correct root number (n) in your calculation.
  • Rounding errors: Be aware that calculators may round results, especially for irrational roots.
  • Negative radicands: Remember that even roots of negative numbers are not real numbers (they are complex).

For example, to calculate the 4th root of -16:

Input: (-16)^(1/4)

Result: 2i (where i is the imaginary unit)

Always double-check your calculations and understand the limitations of your calculator.

Frequently Asked Questions

What is the difference between a square root and an nth root?

A square root is a special case of an nth root where n=2. The square root of a number x is x^(1/2). Other nth roots, like cube roots (n=3) and fourth roots (n=4), follow the same general formula x^(1/n).

Can I calculate roots of numbers without a calculator?

Yes, you can estimate roots using methods like the Babylonian method (also known as Heron's method) or by using logarithms. However, these methods are more time-consuming and less precise than using a calculator.

What happens if I try to calculate the square root of a negative number?

For real numbers, the square root of a negative number is not defined. However, in complex numbers, the square root of a negative number is an imaginary number. For example, √(-1) = i, where i is the imaginary unit.

How do I calculate the nth root of a fraction?

To calculate the nth root of a fraction, you can apply the root to both the numerator and the denominator separately. For example, the cube root of 8/27 is (8/27)^(1/3) = 2/3.