How to Calculate the 5th Root of a Number Manually?

It's difficult to calculate the 5th root of a number precisely manually without advanced techniques or calculators, but there are estimation methods. The Numberphile video you referenced hints at a trick for identifying the 5th root if it's an integer and the number provided is the result of raising that integer to the 5th power. This trick relies on recognizing patterns in the last digit and understanding the relative scale of the 5th power.

Here's a breakdown of a method suitable for perfect fifth powers where the root is an integer. Keep in mind this is not a general method for any number.

1. Understanding the Last Digit Pattern:

The last digit of a number raised to the 5th power follows a predictable pattern. The last digit of the base number is the same as the last digit of the 5th power.

• If the number ends in 0, its 5th power ends in 0.
• If the number ends in 1, its 5th power ends in 1.
• If the number ends in 2, its 5th power ends in 2.
• If the number ends in 3, its 5th power ends in 3.
• If the number ends in 4, its 5th power ends in 4.
• If the number ends in 5, its 5th power ends in 5.
• If the number ends in 6, its 5th power ends in 6.
• If the number ends in 7, its 5th power ends in 7.
• If the number ends in 8, its 5th power ends in 8.
• If the number ends in 9, its 5th power ends in 9.

This makes identifying the last digit of the 5th root easy.

2. Estimating the Magnitude (Number of Digits):

Next, consider the magnitude of the number. You can use benchmarks like:

• 10⁵ = 100,000
• 20⁵ = 3,200,000
• 30⁵ = 24,300,000
• 100⁵ = 10,000,000,000 (10 Billion)

By comparing your target number to these benchmarks, you can estimate how many digits the 5th root has and thus, the tens digit (if any).

3. Combining Last Digit and Magnitude Estimation:

Now combine the last digit determination (step 1) with the magnitude estimation (step 2). For example, if you're given the number 3,276,8.

• The last digit is 8, so the 5th root must end in 8.
• 3,276,8 is between 10⁵ (100,000) and 20⁵ (3,200,000). So the 5th root is likely a two-digit number starting with 1.
• Therefore, the 5th root is most likely 18. Checking, 18⁵ = 1,889,568. It's not correct.
• Given that 3,276,8 is lower than 20⁵ and higher than 10⁵, let's try a number below 20 that end in 8. Let's try 8. 8⁵ = 32,768.

Important Considerations:

• This method only works if the number provided is a perfect 5th power of an integer. It's not a general solution for finding 5th roots of arbitrary numbers.
• For non-perfect 5th powers, you'd need more advanced numerical methods like the Newton-Raphson method (which typically requires a calculator or computer).
• For larger numbers, magnitude estimation becomes trickier, and this method becomes less practical.

In summary, the "trick" leverages the predictable last-digit pattern and benchmarks to quickly identify the 5th root if it's an integer. It's a neat demonstration, but it's not a substitute for more general root-finding techniques.