Michael Daniels

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Instant Mental Calculation of Fifth Roots

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Learn how to mentally calculate fifth roots and amaze your friends!



By learning this simple system you will be able to instantly calculate the fifth root of the spectator's number.

Ask the spectator to choose any whole number less than 100 and, using a calculator, to find its fifth power by multiplying the number by itself, then multiplying the answer by the original number another three times.

Alternatively, if the spectator knows how to use scientific calculator, they can enter the original number, then press the "x^y" key followed by 5 and =.

The spectator then calls out the answer, and you instantly reveal the original number (i.e., the fifth root).

Note that this method will not work with fifth powers of numbers greater than 99. Also the method only works when the fifth root is a natural number (integer).


Step 1: Learn the Fifth Powers of 0 to 9

To master the system you must learn by heart the fifth powers of numbers 0 to 9, which are shown in the table below. You also need to consider the last digit of each fifth power.

Number Fifth PowerLast Digit
0 00
3243 3
5 31255
832768 8

Note how the last digit of every fifth power ends with the original number.

Step 2: Determine the Fifth Root

Ignore the last five digits of the number called out by the spectator and choose the memorised fifth power which is just lower (or equal) to the remaining number. The corresponding fifth root is the first digit of your answer.

Now consider the last digit of the number called out by the spectator. This is the last digit of your answer (see table above).


Spectator CallsIgnore last 5Lower 5th Power First DigitLast DigitFifth Root
2488322112 12
17210368172322 828
69343957693243 3737
2824752492824 10244949
418195493 418131255353
7776000007776777660 60
18042293511804216807 7171
470427017647042 327688686
7339040224 73390590499494

Step 3: Practice, Practice, Practice!

Before you try this out on your friends, you should practice until you can calculate the fifth root instantly and without error. If you can't do this, then practice some more! To practice and assess your ability, you can use the test below.




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