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How to Take The Third Root on A Calculator

Reviewed by Calculator Editorial Team

Calculating cube roots is a fundamental mathematical operation that finds applications in geometry, algebra, and real-world measurements. Whether you're solving equations, analyzing three-dimensional shapes, or working with volumes, understanding how to find cube roots is essential. This guide explains how to take the third root on a calculator, including different methods, formulas, and practical examples.

How to Calculate Cube Roots

The cube root of a number is a value that, when multiplied by itself three times, gives the original number. For example, the cube root of 27 is 3 because 3 × 3 × 3 = 27.

Most modern calculators have a built-in cube root function, but understanding how to use it properly is key. Here's a step-by-step guide:

  1. Enter the number you want to find the cube root of.
  2. Locate the cube root function on your calculator. This is often labeled as "x³" or "³√x".
  3. Press the cube root button.
  4. Review the result displayed on the calculator screen.

Note: If your calculator doesn't have a dedicated cube root button, you can use the exponent function (yˣ) by entering 1/3 as the exponent.

Different Calculator Methods

Calculators offer several ways to find cube roots, depending on the model and features available. Here are the most common methods:

1. Dedicated Cube Root Button

Many scientific calculators have a specific button labeled "³√x" or "x³". This is the most straightforward method.

2. Exponent Function

If your calculator lacks a dedicated cube root button, you can use the exponent function (yˣ) by entering 1/3 as the exponent. For example, to find the cube root of 64, you would enter 64^(1/3).

3. Logarithmic Approach

For calculators without cube root functions, you can use logarithms. The formula is: ³√x = 10^(log(x)/3). This method is less common but demonstrates the mathematical relationship between cube roots and logarithms.

The Cube Root Formula

The cube root of a number x can be expressed mathematically as:

³√x = y, where y × y × y = x

This formula shows that the cube root of x is a number y that, when multiplied by itself three times, equals x. For example:

  • ³√8 = 2, because 2 × 2 × 2 = 8
  • ³√125 = 5, because 5 × 5 × 5 = 125
  • ³√-27 = -3, because (-3) × (-3) × (-3) = -27

Worked Examples

Let's look at some practical examples of calculating cube roots using different methods.

Example 1: Using a Dedicated Cube Root Button

Find the cube root of 512 using a calculator with a dedicated cube root button.

  1. Press the "512" button.
  2. Press the "³√x" button.
  3. The calculator displays "8" as the result.

Verification: 8 × 8 × 8 = 512, so the calculation is correct.

Example 2: Using the Exponent Function

Find the cube root of 216 using the exponent function.

  1. Press the "216" button.
  2. Press the "yˣ" button.
  3. Enter "1/3" as the exponent.
  4. The calculator displays "6" as the result.

Verification: 6 × 6 × 6 = 216, confirming the calculation.

Common Mistakes

When calculating cube roots, several common mistakes can lead to incorrect results. Here are some pitfalls to avoid:

1. Confusing Square Roots with Cube Roots

Square roots (²√x) and cube roots (³√x) are different operations. A square root is a number that, when multiplied by itself twice, gives the original number, while a cube root is a number that, when multiplied by itself three times, gives the original number.

2. Forgetting Negative Roots

Cube roots can be negative. For example, the cube root of -8 is -2, because (-2) × (-2) × (-2) = -8. Forgetting this can lead to incorrect results when dealing with negative numbers.

3. Using the Wrong Function

Using the square root function instead of the cube root function will give incorrect results. Always ensure you're using the correct function for the operation you need to perform.

Frequently Asked Questions

What is the difference between a square root and a cube root?
A square root is a number that, when multiplied by itself twice, gives the original number. A cube root is a number that, when multiplied by itself three times, gives the original number. For example, ²√16 = 4 (4 × 4 = 16), while ³√27 = 3 (3 × 3 × 3 = 27).
Can I find the cube root of a negative number?
Yes, you can find the cube root of a negative number. The result will also be negative. For example, ³√(-8) = -2, because (-2) × (-2) × (-2) = -8.
How do I calculate the cube root of a fraction?
To calculate the cube root of a fraction, you can use the same methods as for whole numbers. For example, ³√(1/8) = 1/2, because (1/2) × (1/2) × (1/2) = 1/8.
What if my calculator doesn't have a cube root button?
If your calculator lacks a dedicated cube root button, you can use the exponent function by entering 1/3 as the exponent. For example, to find ³√64, enter 64^(1/3).