# Instant Mental Calculation of Fifth Roots

## Fifth Root Calculation

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

Following a few simple steps 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 cube by multiplying the number by itself, then multiplying the answer by the original number.

Alternatively, if the spectator knows how to use scientific calculator, they can enter the original number, then press the x^y key followed by 3 and =.

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

Note that this method will NOT work with non-whole numbers, or with numbers greater than 99.

## 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.

Number5th PowerLast Digit
000
111
2322
32433
410244
531255
677766
7168077
8327688
9590499

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).

#### Example 1

Spectator calls 1073741824

• Ignore last five digits = 10737
• Lower (or equal) fifth power = 7776
• Fifth root of 7776 is 6 = first digit of answer
• Last digit of spectator's call is 4 = last digit of answer
• Therefore the fifth root of 1073741824 is 64

#### Example 2

Spectator calls 130691232

• Ignore last five digits = 1306
• Lower (or equal) fifth power = 1024
• Fifth root of 1024 is 4 = first digit of answer
• Last digit of spectator's call is 2 = last digit of answer
• Therefore the fifth root of 130691232 is 42

## Further Examples

Called
Fifth
Power
Ignore
Last
Five
Lower
(or =)
Power
First
Digit
(F)
Last
Digit
of Call
Last
Digit
(S)
Fifth
Root
= FS
2488322112212
172103681723228828
6934395769324337737
2824752492824102449949
4181954934181312553353
7776000007776777660060
1804229351180421680771171
4704270176470423276886686
7339040224733905904994494

## 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 Fifth Root Tester shown below.