Doing Cube Roots in Your Head
For this activity, you will have to know the cubes of the counting numbers from 1 to 9.
Activity
(Students should have calculators.)
Ask students to choose a two-digit number. They should write their choices down so
they donÕt forget. Now the students should (individually) cube their numbers, using their
calculators. When they tell you their results, you will tell them the numbers they started
with.
The Technique
There are two separate processes to obtain the two digits and thereby reconstruct the
studentÕs original number.
For the ones digit, cube the ones digit of the number given to you by the student. The
ones digit of this result is the ones digit of the number he or she started with. For
example, if the studentÕs cube is 103,823, then the ones digit of the original number is the
ones digit of 3 cubed, 7 (the ones digit of 27).
For the tens digit, you must look at the digits to the left of the comma in the studentÕs
cube (there will be 1, 2 or 3 digits here; in our example we look at 103). Now we find
where 103 falls on the list of perfect cubes of the numbers between 1 and 9, and the digits
that gave those cubes. 103 is between 64 and 125, which are 4-cubed and 5-cubed,
respectively. The smaller of the two is 4, so that is the tens digit of the studentÕs original
number.
Thus the studentÕs original number is 47.
Special case: If the thousands digits equals a perfect cube
(say 125), the tens digit is the cube root of that number (5).
YouÕll notice that the ones digit of the cube is often also the
same as the ones digit of the original number. This occurs
with numbers that end in 0,1,4,5,6, and 9. The rest of the
time the ones digit of the cube root is ten minus the ones digit
of the cube (as in our two examples).
Example
3
Suppose N = 12,167.
Then the ones digit of N is 3
since 73 = 343 ends in 3.
The tens digit of N is 2 since
23 = 8 ² 12 < 27 = 33.
So, N = 23 and 233 = 12,167.
The general idea for finding the tens digit, by using the ÒthousandsÓ part of the cube, say
t, is to locate two perfect cubes s3 and u3, so that s3 ² t < u3; then s is the tens digit of the
studentÕs original number.
Why it Works
For the units digit, it is always true that the ones digit of a is the same as the ones digit of
a9. So when the student presents you with a3, and you cube the ones digit of a3, you will
have the ones digit of (a3)3 = a9, which is the same as the ones digit of a.
For the tens digit, let y be the tens digit of the studentÕs original number. Then y falls
between two multiples of 10, x and z (possibly equaling x). Note that x and y have the
same tens digit. So x3 ² y3 < z3. Since x3 and z3 both end in 000 and since you know the
cubes of the numbers from 1 to 9, you know the digits past the hundreds place of both x3
and z3. Then you know the interval that y3 falls in. Since y3 is smaller than z3 and no
smaller than x3, the tens digit of y is now knownÑthe same as the tens digit of x.
Note that this method works only with perfect cubes.
Assignments:
1. How would you do cubes of three-digit numbers?
2. Put the integers from 1 to 9 and their cubes on the board and have the students do
the trick themselves (perhaps in pairs).
3. Some numbers, like 99, will appear more often than others when students choose
their original numbers. Why is that?
4. Would this trick work with 4th powers or 5th powers? Why or why not?
5. What if the students started with three-digit numbers? What would be the range
of their cubes? Would the units digit part of the trick change? What would you
have to know to get the remaining digits of the students' original three-digit
numbers?