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How to Take The 3rd Root on A Calculator

Reviewed by Calculator Editorial Team

Calculating the cube root of a number is a fundamental mathematical operation with applications in geometry, algebra, and engineering. This guide explains how to find the cube root using both calculator methods and manual techniques, along with practical examples and common uses.

How to Calculate the Cube Root

The cube root of a number \( x \) is a value that, when multiplied by itself three times, gives the original number. In mathematical terms:

If \( y = \sqrt[3]{x} \), then \( y^3 = x \).

For example, the cube root of 27 is 3 because \( 3 \times 3 \times 3 = 27 \).

Using a Calculator

Most scientific calculators have a dedicated cube root function. 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 cube root of.
  3. Press the cube root function button (often labeled as \( \sqrt[3]{x} \) or \( x^{1/3} \)).
  4. Press the equals (=) button to display the result.

Using a Computer or Smartphone

If you don't have a scientific calculator, you can use the cube root function on your computer or smartphone:

  1. Open a calculator app that supports scientific functions.
  2. Enter the number you want to find the cube root of.
  3. Press the cube root function button (often labeled as \( \sqrt[3]{x} \) or \( x^{1/3} \)).
  4. Press the equals (=) button to display the result.

Note: If your calculator doesn't have a dedicated cube root function, you can calculate it by raising the number to the power of 1/3.

Calculator Methods

There are several methods to calculate the cube root using a calculator:

Direct Cube Root Function

The most straightforward method is to use the calculator's built-in cube root function. This is available on most scientific calculators and computer applications.

Exponentiation Method

If your calculator doesn't have a dedicated cube root function, you can calculate it by raising the number to the power of 1/3.

\( \sqrt[3]{x} = x^{1/3} \)

Step-by-Step Example

Let's find the cube root of 64 using the exponentiation method:

  1. Enter the number 64 on your calculator.
  2. Press the exponentiation button (often labeled as \( x^y \)).
  3. Enter 1/3 as the exponent.
  4. Press the equals (=) button to display the result.

The result should be 4, since \( 4 \times 4 \times 4 = 64 \).

Manual Calculation Methods

If you don't have access to a calculator, you can estimate the cube root using manual methods:

Prime Factorization Method

This method involves breaking down the number into its prime factors and then grouping them into triplets.

Example: Find the cube root of 512.

  1. Factorize 512: \( 512 = 2 \times 256 = 2 \times 2 \times 128 = \ldots = 2^9 \).
  2. Group the prime factors into triplets: \( (2 \times 2 \times 2) \times (2 \times 2 \times 2) \times (2 \times 2 \times 2) \).
  3. Take one factor from each triplet: \( 2 \times 2 \times 2 = 8 \).

Therefore, \( \sqrt[3]{512} = 8 \).

Estimation Method

For numbers that aren't perfect cubes, you can estimate the cube root by finding two perfect cubes that the number lies between.

Example: Find the cube root of 28.

  1. Find perfect cubes near 28: \( 3^3 = 27 \) and \( 4^3 = 64 \).
  2. Since 28 is closer to 27 than to 64, the cube root is approximately 3.04.

Common Uses of Cube Roots

Cube roots have several practical applications in various fields:

Geometry

Cube roots are used to find the volume of a cube when the side length is known. The formula for the volume of a cube is:

Volume = \( \text{side length}^3 \)

Algebra

Cube roots are used to solve equations involving cubes, such as \( x^3 = 8 \), which has the solution \( x = \sqrt[3]{8} = 2 \).

Engineering

Cube roots are used in engineering calculations involving cubic dimensions, such as calculating the side length of a cube given its volume.

Frequently Asked Questions

What is the difference between a square root and a cube root?

A square root of a number \( x \) is a value that, when multiplied by itself, gives \( x \). A cube root is a value that, when multiplied by itself three times, gives \( x \).

How do I find the cube root of a negative number?

The cube root of a negative number is negative. For example, the cube root of -8 is -2 because \( (-2) \times (-2) \times (-2) = -8 \).

Can I find the cube root of a fraction?

Yes, you can find the cube root of a fraction by finding the cube roots of the numerator and the denominator separately. For example, \( \sqrt[3]{\frac{1}{8}} = \frac{\sqrt[3]{1}}{\sqrt[3]{8}} = \frac{1}{2} \).

What is the cube root of 1?

The cube root of 1 is 1 because \( 1 \times 1 \times 1 = 1 \).