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How to Find Third Root on Calculator

Reviewed by Calculator Editorial Team

Finding the third root of a number is a fundamental mathematical operation that has practical applications in various fields. This guide explains how to find the third root using a calculator, including step-by-step instructions, real-world examples, and common pitfalls to avoid.

What is a Third Root?

The third root of a number is a value that, when multiplied by itself three times, gives the original number. Mathematically, the third root of a number \( x \) is a number \( y \) such that:

\( y^3 = x \)

For example, the third root of 27 is 3 because \( 3 \times 3 \times 3 = 27 \). The third root is also known as the cube root.

How to Calculate Third Root

There are two primary methods to find the third root of a number: using a calculator and manual calculation. Both methods are explained in detail below.

Using a Calculator

Most scientific and graphing calculators have a built-in function to find the third root of a number. Here's how to use it:

  1. Turn on your calculator and ensure it's in the appropriate mode (usually "DEG" or "RAD" for scientific calculators).
  2. Enter the number for which you want to find the third root.
  3. Press the "y^x" or "x^y" key to raise the number to the power of 1/3.
  4. Press the "=" key to calculate the result.

For example, to find the third root of 64:

  1. Enter 64 on your calculator.
  2. Press the "y^x" key.
  3. Enter 1/3.
  4. Press "=" to get the result: 4.

Note: Some calculators may have a dedicated "cube root" function. If available, use it for faster results.

Manual Calculation

If you don't have a calculator, you can estimate the third root using the following steps:

  1. Find the square root of the number using the manual method.
  2. Take the square root of the result from step 1 to find the cube root.

For example, to find the third root of 27:

  1. Find the square root of 27, which is approximately 5.196.
  2. Find the square root of 5.196, which is approximately 2.279.

This manual method provides an estimate, but it's less precise than using a calculator.

Real-World Examples

The third root has practical applications in various fields. Here are a few examples:

  • Volume Calculations: The third root is used to find the side length of a cube when the volume is known.
  • Physics: The third root appears in equations related to energy and work.
  • Engineering: It's used in calculations involving three-dimensional shapes and structures.

For instance, if a cube has a volume of 125 cubic units, the side length can be found by taking the third root of 125, which is 5 units.

Common Mistakes

When finding the third root, it's easy to make the following mistakes:

  • Confusing with Square Root: The third root is different from the square root. Ensure you're using the correct function on your calculator.
  • Incorrect Exponent: Remember that the third root is equivalent to raising the number to the power of 1/3, not 1/2.
  • Rounding Errors: Be cautious when rounding intermediate results, especially in manual calculations.

FAQ

What is the difference between square root and third root?

The square root of a number is a value that, when multiplied by itself, gives the original number. The third root is a value that, when multiplied by itself three times, gives the original number.

Can I find the third root of a negative number?

Yes, you can find the third root of a negative number. For example, the third root of -8 is -2 because \( (-2) \times (-2) \times (-2) = -8 \).

How do I find the third root of a fraction?

To find the third root of a fraction, take the third root of the numerator and the denominator separately. For example, the third root of 8/27 is 2/3 because \( (2/3)^3 = 8/27 \).