Numberphile has a nice video on the fifth root trick: someone raises a two-digit number to the 5th power, reads the number aloud, and you tell them immediately what the number was.

Here’s the trick in a nutshell. For any number *n*, *n*^{5} ends in the same last digit as *n*. You could prove that by brute force or by Euler’s theorem. So when someone tells you *n*^{5}, you immediately know the last digit. Now you need to find the first digit, and you can do that by learning, approximately, the powers (10k)^{5} for *i* = 1, 2, 3, …, 9. Then you can determine the first digit by the range.

Here’s where the video is a little vague. It says that you don’t need to know the powers of 10*k* very accurately. This is true, but just how accurately *do* you need to know the ranges?

If the two-digit number is a multiple of 10, you’ll recognize the zeros at the end, and the last non-zero digit is the first digit of *n*. For example, if *n*^{5} = 777,600,000 then you know *n* is a multiple of 10, and since the last non-zero digit is 6, *n* = 60.

So you need to know the fifth powers of multiples of 10 well enough to distinguish (10*k* – 1)^{5} from (10*k* + 1)^{5}. The following table shows what these numbers are.

|---+---------------+---------------| | k | (10k - 1)^5 | (10k + 1)^5 | |---+---------------+---------------| | 1 | 59,049 | 161,051 | | 2 | 2,476,099 | 4,084,101 | | 3 | 20,511,149 | 28,629,151 | | 4 | 90,224,199 | 115,856,201 | | 5 | 282,475,249 | 345,025,251 | | 6 | 714,924,299 | 844,596,301 | | 7 | 1,564,031,349 | 1,804,229,351 | | 8 | 3,077,056,399 | 3,486,784,401 | | 9 | 5,584,059,449 | 6,240,321,451 | |---+---------------+---------------|

So any number less than a million has first digit 1. Any number between 1 million and 3 million has first digit 2. Etc.

You could choose the following boundaries, if you like.

|---+----------------| | k | upper boundary | |---+----------------| | 1 | 1,000,000 | | 2 | 3,000,000 | | 3 | 25,000,000 | | 4 | 100,000,000 | | 5 | 300,000,000 | | 6 | 800,000,000 | | 7 | 1,700,000,000 | | 8 | 3,200,000,000 | | 9 | 6,000,000,000 | |---+----------------|

The Numberphile video says you should have someone say the number aloud, in words. So as soon as you hear “six billion …”, you know the first digit of *n* is 9. If you hear “five billion” or “four billion” you know the first digit is 8. If you hear “three billion” then you know to pay attention to the next number, to decide whether the first digit is 7 or 8. Once you hear the first few syllables of the number, you can stop pay attention until you hear the last syllable or two.

I think there is a better and more generally useful way to find the first digits: take logs base 10. Because log 2 is 0.3 to remarkable accuracy you can estimate logs quickly. Eg the number in the video is 79 million and some, so it’s log is a little bit less than 7.9. The log of its fifth root is a bit less than 1.6, so it must be thirty something.

Jim, I agree that logarithms would be more reusable knowledge. :)

“you immediately know the first digit.”

–>

“you immediately know the last digit.”

The upper boundary for k=1 is 100,000 instead of 1,000,000, right? This is also easy to remember because 10^5 is 100,000.

No, 100,000 would not work as an upper bound for k = 1 because 19^5 = 161,051. The upper bound has to be bigger than that, and less than 21^5 = 2,476,099. One million is a convenient number between 161,051 and 2,476,099.