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How to Find the nth Root of a Number Using a Calculator?

Published in Calculator Operations 3 mins read

To find the nth root of a number using a calculator, you typically use the yx or xy key and its reciprocal functionality. Here's how to do it:

Understanding the Process

The nth root of a number, let's say 'a', is a value that, when multiplied by itself 'n' times, equals 'a'. Mathematically, this is represented as n√a. This can also be expressed as a(1/n). Calculators use this exponent form.

Steps Using a Standard Calculator

While specific key placement may slightly vary by calculator model, most follow a general pattern:

  1. Enter the Base Number: Type the number you want to find the root of (a).
  2. Press the Power Key: Look for a key labeled as yx, xy, or sometimes ^.
  3. Enter the Reciprocal of the Root: Input the value of 1/n. If you are finding the cube root, you would need to enter 1/3. If you are finding the 4th root, enter 1/4 and so on. Most calculators will require you to perform the division first.
    • Divide 1 by n: Enter 1 ÷ n then press =. The result will be the decimal equivalent of 1/n.
  4. Enter the Power Value: Enter the decimal value you obtained from the division or press the 1/x button.
  5. Press the Equal Sign: Press the = key to get your final answer.

Example: Finding the Cube Root of 8 (3√8)

  1. Enter the Base: Type 8
  2. Press the Power Key: Press yx
  3. Enter 1 divided by n (3): Type 1 ÷ 3 =. You should see 0.333333333 on the display.
  4. Enter the power value: You can now enter 0.333333333 if your calculator does not have a way to keep the value from the previous equation in use or just press the = sign again.

The final result should be 2, meaning the cube root of 8 is 2.

Using Online Calculators

Many websites provide online root calculators if you don't have a physical one. Usually, these are more user-friendly, with dedicated fields for the root and number, eliminating the need to calculate reciprocals manually.

Alternative Approach on Some Calculators (Using the Video Reference)

The video reference describes an alternative approach that involves iterative calculations, which can be done on basic calculators that do not have the yx button. This method is less efficient and accurate and relies on repeated operations:

  1. Start with the Number: Enter the number you want to find the root of (a).
  2. Press Square Root: Press the square root button.
  3. Add 1: Add 1 and press the equal sign.
  4. Multiply: Press the multiplication sign, then press the equal sign.
  5. Repeat: Repeat step 3 and 4, and watch the number tend to 1.
  • Note: This process does not provide a precise result for nth roots but attempts to get close.

Important Considerations:

  • Fractional exponents: Make sure you understand how fractional exponents relate to roots.
  • Calculator functions: Be familiar with your calculator’s specific keys and their functions.
  • Precision: Be aware that calculators may use approximations, especially with irrational roots.

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