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How to Get Sixth Root on Graphing Calculator

Reviewed by Calculator Editorial Team

Calculating the sixth root of a number is a common mathematical operation that can be performed on most graphing calculators. This guide explains how to find the sixth root using a graphing calculator, including step-by-step instructions and practical examples.

Introduction

The sixth root of a number x is a value that, when raised to the power of 6, equals x. Mathematically, the sixth root of x is represented as x^(1/6). This operation is useful in various mathematical and scientific applications, including solving equations and analyzing data.

Most modern graphing calculators, such as the TI-84, TI-89, and Casio fx-CG50, have built-in functions to calculate roots. This guide will walk you through the process of finding the sixth root using these calculators.

Sixth Root Formula

The sixth root of a number x can be expressed using the following formula:

x^(1/6) = y

where y is the sixth root of x.

This formula states that the sixth root of x is the number y that, when multiplied by itself six times, equals x.

Using the Calculator

Step-by-Step Instructions

  1. Turn on your graphing calculator and ensure it is in the appropriate mode (e.g., math mode).
  2. Press the appropriate key to access the exponent or root function. On most calculators, this is the "x^y" key.
  3. Enter the number for which you want to find the sixth root.
  4. Press the "x^y" key again.
  5. Enter the exponent value of 1/6.
  6. Press the equals (=) key to calculate the result.

Note: The exact key sequence may vary slightly depending on your calculator model. Refer to your calculator's manual for specific instructions.

Manual Calculation

If you need to calculate the sixth root without a calculator, you can use the following steps:

  1. Find the square root of the number. This is the second root.
  2. Take the square root of the result from step 1 to find the fourth root.
  3. Take the square root of the result from step 2 to find the sixth root.

This method is less precise than using a calculator but can be useful for quick estimates.

Examples

Example 1: Finding the Sixth Root of 64

Using the formula x^(1/6), we can find the sixth root of 64 as follows:

64^(1/6) = y

y = 2, because 2^6 = 64.

Therefore, the sixth root of 64 is 2.

Example 2: Finding the Sixth Root of 16,796,160

Using the formula x^(1/6), we can find the sixth root of 16,796,160 as follows:

16,796,160^(1/6) = y

y = 10, because 10^6 = 1,000,000 and 10^6 = 1,000,000.

Therefore, the sixth root of 16,796,160 is 10.

FAQ

What is the difference between a sixth root and a cube root?
The sixth root of a number is the value that, when raised to the power of 6, equals the original number. The cube root is the value that, when raised to the power of 3, equals the original number. Essentially, the sixth root is a more precise measure of the size of a number compared to the cube root.
Can I use a graphing calculator to find the sixth root of a negative number?
No, you cannot find the sixth root of a negative number using real numbers. The sixth root of a negative number is not a real number; it is an imaginary number. Most graphing calculators will display an error message if you attempt to find the sixth root of a negative number.
How do I interpret the result of a sixth root calculation?
The result of a sixth root calculation represents the size of the original number in a more manageable form. For example, the sixth root of 64 is 2, which is easier to work with than the original number 64. The sixth root can be used in various mathematical and scientific applications, such as solving equations and analyzing data.