HomePage RecentChanges

FifthRoots

Introduction

Taking a number to the fifth power means multiplying a number times itself four more times. 5^5^ would be 5 * 5 * 5 * 5 * 5 or 3,125. The fifth root of 3,125 would therefore be 5. In this article, you will learn how, given a number that has been taken to the fifth power, to extract that number's fifth root mentally.

Starting the Routine

Provide your audience member with a calculator that can display 10 digits, and ask them to enter a number from 1 through 99 (The highest number you'll be able to see the fifth power of on an 8-digit calculator is 39). If the calculator has a button marked x^n^, tell them to press that button, and then press 5, otherwise instruct them on how to multiply the number times itself four more times. Emphasize that you don't want to see the calculator at any point. Ask them to read the number aloud. You may wish to write the problem on a notepad or board to help the audience visualize the apparent difficulty of this feat, as well as give yourself extra time to calculate the answer.

First, you will calculate the ones digit of the fifth root, followed by the tens digit of the fifth root, thus giving you the complete fifth root.

Getting the Ones Digit

Getting the ones digit of the answer is easy. The ones digit will always be the same as the last digit of the given number. If the number ends in 1, the ones digit of the fifth root will be 1. If the number ends in 2, the ones digit of the fifth root will be 2, and so on.

Getting the Tens Digit

Calculating the tens digit of the fifth root is a little trickier.

If the given number itself is less than 100,000 (10^5^), then there is no number in the tens digit, and the fifth root is merely whatever digit is in the ones place of the answer. If you're given the number 7,776, you can instantly say that the fifth root of the number is 6 (the ones digit in the given number).

If the given number is greater than 100,000, you ignore the ones digit now, as well as the four numbers to the left of the ones digit (the numbers in the tens, hundreds, thousands, and ten-thousands places). You're only going to be concerned with numbers no smaller than the hundred-thousands.

Let's say you're given the number 130,691,232. The ones digit of the fifth root is, obviously, 2. Ignoring the ones digit, and the four digits to the left of it, we're left with 1306. This is the part of the number that will be used to find the tens digit.

You'll need to memorize some number ranges in order to determine the tens digit, though:

If the remaining number is in the range The tens digit will be
1 - 30 1
30 - 230 2
230 - 1,000 3
1,000 - 3,000 4
3,000 - 7,500 5
7,500 - 16,000 6
16,000 - 32,000 7
32,000 - 57,000 8
57,000 - 99,000 9

These ranges are specially chosen so that no given number will fall right on them.

Those experienced with the Major System can use it to link the numbers this way:

1 1
30 2
230 3
1,000 4
3,000 5
7,500 6
16,000 7
32,000 8
57,000 9

Each number, would of course be broken down into appropriate words or phrases and then linked to the appropriate peg.

Once you've linked the ranges, you simply count to the highest range you can without going over the number. With our example, you'd think of 1306 and realize that it is greater than the number associated with 4, but lower than 5. So we still get 4 as the tens digit. Since the last digit in 130,691,232 is a 2, this gives us 42 as the fifth root.

Examples

To help understand this process, I'll provide a few more examples.

Our first example is 59,049. The last digit is 9, and it's less than 100,000, so the fifth root of 59,049 is 9.

Our next example is 6,436,343. The last digit is 3, so the fifth root of the ones digit is 3. Ignoring that and the next four number to the left of it, we're left with 64. Using the linked system we realize that 64 is more than 30 (range 2) but less than 230 (range 3), so the tens digit must be 2. This gives us 23.

Our final example will be 1,934,917,632. The ones digit will obviously be 2. Ignoring that and the four numbers to the left, we're left with 19,349. This is greater than 16,000 but less than 32,000, so the tens digit is 7. This gives us 72 as the fifth root.

Advanced Version

If you're willing to put in a little extra practice, and supply your audience member with a 12-digit calculator, you can add an extra sequence to really impress them!

With a 12-digit calculator, your audience can clearly calculate fifth powers up to, and including, 251. To be able to answer numbers from 100 to 251, the method is similar, but you'll be solving for the hundreds and tens places at the same time.

To get the appropriate hundreds and tens digit, you're going to ignore the ones place of the given number and the 8 digits immediately to its left. For those in the U.S. and/or the scientific community, you're only going to be concerned with the billions. For those in countries that use a different system, you'll only be focusing on the numbers in the milliards.

Here are the ranges for the numbers from 100 through 251:

If the remaining number is in the range The hundreds and tens digit will be
10 - 16 10
16 - 24 11
24 - 36 12
36 - 52 13
52 - 75 14
75 - 104 15
104 - 140 16
140 - 188 17
188 - 244 18
244 - 320 19
320 - 400 20
400 - 510 21
510 - 630 22
630 - 790 23
790 - 970 24
970 - 999 25

So, for linking purposes, here are the numbers you have to associate:

link this number to this number
10 10
16 11
24 12
36 13
52 14
75 15
104 16
140 17
188 18
244 19
320 20
400 21
510 22
630 23
790 24
970 25

Just as before, each number is broken down into appropriate words or phrases and then linked to the appropriate peg. Also, note that this chart won't interfere with your links for just the tens digits taught earlier.

Using this knowledge, let's go through a few examples:

Our first example will be 48,261,724,457. Since the number is greater than 10 billion/10 milliard, we know that the fifth root will be greater than 100. We also know that the ones digit of the root will be 7. Ignoring the ones digit and the eight immediately to the left of it, we're left with 48. That's greater than 36 (range 13) but less than 52 (range 14), so the hundreds and tens digit of the fifth root is 13. This gives us 137 as the fifth root.

In this next example, we're working with the number 503,756,397,099. The number 503 is greater than 400, but less than 510, so the hundreds and the tens digit is 21. The final digit of the given number is 9, so the fifth root of 503,756,397,099 is 219.

Our final example is 996,250,626,251. The 996 is greater than 970, so the hundreds and tens places of the fifth root are 25. The last digit is 1, so the ones place of the fifth root is 1. This gives us 251 as the fifth root of 996,250,626,251.

Practice Exercises

Here are some numbers to practice with. If you have problems, your computer or scientific calculator should be able to give you the correct answers.

Numbers from 1 to 99 taken to the fifth power:
3,276,800,000
7,776
459,165,024
20,511,149
371,293
2,887,174,368
844,596,301
9,509,900,499
79,235,168
7,737,809,375
16,807

Numbers from 100 to 251 taken to the fifth power:
10,000,000,000
59,797,108,943
168,874,213,376
312,079,600,999
12,762,815,625
539,218,609,632
101,621,504,799
643,634,300,000
21,924,480,357
957,186,876,249

Other Resources

CubingNumbers
CubeRoots