Cal11 calculator
Search calculator...

How to Plug 5th Root in A Calculator

Reviewed by Calculator Editorial Team

Calculating the 5th root of a number is a common mathematical operation that can be performed using a calculator. This guide explains how to input and compute the 5th root on standard calculators, including scientific and graphing models.

How to Calculate the 5th Root

The 5th root of a number \( x \) is a value that, when raised to the power of 5, equals \( x \). Mathematically, it's represented as \( \sqrt[5]{x} \).

Formula: \( \sqrt[5]{x} = x^{1/5} \)

To calculate the 5th root of a number, you can use either a calculator or perform the calculation manually using logarithms.

Using a Calculator

Most scientific and graphing calculators have a dedicated key or function for calculating roots. Here's how to use it:

  1. Turn on your calculator and clear any previous calculations.
  2. Enter the number you want to find the 5th root of.
  3. Press the "yx" or "xy" key (this varies by calculator model).
  4. Enter "0.2" (since 1/5 = 0.2) or use the reciprocal function if available.
  5. Press the equals (=) key to get the result.

Note: Some calculators may have a dedicated "nth root" function. Check your calculator's manual for specific instructions.

Manual Calculation

If you don't have a calculator, you can use logarithms to find the 5th root:

  1. Take the natural logarithm of the number: \( \ln(x) \).
  2. Divide the result by 5: \( \frac{\ln(x)}{5} \).
  3. Exponentiate the result using the natural exponential function: \( e^{\frac{\ln(x)}{5}} \).

Manual Formula: \( \sqrt[5]{x} = e^{\frac{\ln(x)}{5}} \)

Worked Examples

Example 1: Calculating the 5th Root of 32

Using a calculator:

  1. Enter 32.
  2. Press the "yx" key.
  3. Enter 0.2.
  4. Press equals.

The result is 2, because \( 2^5 = 32 \).

Example 2: Calculating the 5th Root of 1000

Using a calculator:

  1. Enter 1000.
  2. Press the "yx" key.
  3. Enter 0.2.
  4. Press equals.

The result is approximately 3.98, because \( 3.98^5 \approx 1000 \).

Frequently Asked Questions

What is the difference between a square root and a 5th root?
The square root finds a number that, when multiplied by itself, gives the original number. The 5th root finds a number that, when multiplied by itself five times, gives the original number.
Can I calculate the 5th root of a negative number?
Yes, but the result will be a negative number. For example, the 5th root of -32 is -2, because \( (-2)^5 = -32 \).
How accurate are calculator results for roots?
Most calculators provide accurate results to at least 10 decimal places. For more precise calculations, scientific calculators or software may be needed.
Is there a difference between the 5th root and the reciprocal of the 5th power?
No, they are mathematically equivalent. The 5th root of a number \( x \) is the same as \( x \) raised to the power of 1/5.