Points to
remember for speedy calculation of cube roots of perfect cubes
1. To calculate cube root of any
perfect cube quickly, we need to remember the cubes of 1 to 10 which is given
below.
|
1³
|
=
|
1
|
|
2³
|
=
|
8
|
|
3³
|
=
|
27
|
|
4³
|
=
|
64
|
|
5³
|
=
|
125
|
|
6³
|
=
|
216
|
|
7³
|
=
|
343
|
|
8³
|
=
|
512
|
|
9³
|
=
|
729
|
|
10³
|
=
|
1000
|
2. From
the above cubes of 1 to 10, we need to remember an interesting property.
|
1³
|
=
|
1
|
=>
|
If
the last digit of the perfect cube = 1, the last digit of the cube root = 1
|
|
2³
|
=
|
8
|
=>
|
If
the last digit of the perfect cube = 8, the last digit of the cube root = 2
|
|
3³
|
=
|
27
|
=>
|
If
the last digit of the perfect cube = 7, the last digit of the cube root = 3
|
|
4³
|
=
|
64
|
=>
|
If
the last digit of the perfect cube = 4, the last digit of the cube root = 4
|
|
5³
|
=
|
125
|
=>
|
If
the last digit of the perfect cube =5, the last digit of the cube root = 5
|
|
6³
|
=
|
216
|
=>
|
If
the last digit of the perfect cube = 6, the last digit of the cube root = 6
|
|
7³
|
=
|
343
|
=>
|
If
the last digit of the perfect cube = 3, the last digit of the cube root = 7
|
|
8³
|
=
|
512
|
=>
|
If
the last digit of the perfect cube = 2, the last digit of the cube root = 8
|
|
9³
|
=
|
729
|
=>
|
If
the last digit of the perfect cube = 9, the last digit of the cube root = 9
|
|
10³
|
=
|
1000
|
=>
|
If
the last digit of the perfect cube = 0, the last digit of the cube root = 0
|
It’s very
easy to remember the relations given above because
|
1
|
->
|
1
|
(Same
numbers)
|
|
8
|
->
|
2
|
(10’s
complement of 8 is 2 and 8+2 = 10)
|
|
7
|
->
|
3
|
(10’s
complement of 7 is 3 and 7+3 = 10)
|
|
4
|
->
|
4
|
(Same
numbers)
|
|
5
|
->
|
5
|
(Same
numbers)
|
|
6
|
->
|
6
|
(Same
numbers)
|
|
3
|
->
|
7
|
(10’s
complement of 2 is 7 and 3+7 = 10)
|
|
2
|
->
|
8
|
(10’s
complement of 2 is 8 and 2+8 = 10)
|
|
9
|
->
|
9
|
(Same
numbers)
|
|
0
|
->
|
0
|
(Same
numbers)
|
Also
See
8 -> 2 and 2 -> 8
7 -> 3 and 3-> 7
If we
observe the properties of numbers, Mathematics will be a very interesting
subject and easy to learn.Now let’s see how we can actually find out the cube
roots of perfect cubes very fast.
Example
1: Find Cube Root of 4913
Step 1:
Identify
the last three digits and make groups of three three digits from right side.
That is 4913 can be written as
4, 913
Step 2
Take the
last group which is 913. The last digit of 913 is 3.
Remember point 2, If the
last digit of the perfect cube = 3, the last digit of the cube root = 7
Hence
the right most digit of the cube root = 7
Step 3
Take the
next group which is 4 .
Find out
which maximum cube we can subtract from 4 such that the result >= 0.
We can subtract 1³ = 1 from 4 because 4 - 1 = 3 (If we subtract 2³ = 8 from 4,
4 – 8 = -4 which is < 0)
Hence
the left neighbor digit of the answer = 1.
That
is , the answer = 17
Example
2: Find Cube Root of 804357
Step 1:
Identify
the last three digits and make groups of three three digits from right side.
That is 804357 can be written as
804, 357
Step 2
Take the
last group which is 357. The last digit of 357 is 7.
Remember point 2, If the
last digit of the perfect cube = 7, the last digit of the cube root = 3
Hence
the right most digit of the cube root = 3
Step 3
Take the
next group which is 804 . Find out which maximum cube we can subtract from 4
such that the result >= 0. We can subtract 9³ = 729 from 804
because 804 - 729 = 75 (If we subtract 10³ = 1000 from
729 , 729 – 1000 = -271 which is <
0)
Hence
the left neighbor digit of the answer = 9
That
is , the answer = 93
Perfect Square root
detection
The perfect squares are the squares of the whole numbers such as 1, 4, 9, 16,
25, 36, 49, 64, 81, 100 so on and so forth.
Before learning the procedure, it is wise that the performer memorizes the
squares of the numbers 1-10 which is very elementary:
|
12 = 1
22 = 4
32 = 9
42 = 16
52 = 25
|
62 = 36
72 = 49
82 = 64
92 = 81
102 = 100
|
To extract the square root of any perfect square
follow:
Step-1: Look at the
magnitude of the “hundreds number” (the numbers preceding the last two digits)
and find the largest square that is equal to or less than the number. This
is the 1st part of the answer.
Step-2: Now,
look at the last (unit’s) digit of the number. If the number ends in a:
0 -> then the ending digit of the answer is a 0
1 -> then the
ending digit of the answer is 1 or 9.
4 -> then the
ending digit of the answer is 2 or 8.
5 -> then the
ending digit of the answer is a 5.
6 -> then the
ending digit of the answer is 4 or 6.
9 -> then the
ending digit of the answer is 3 or 7.
To determine the right answer from 2 possible
answers (other than 0 and 5), mentally multiply the findings in step-1 with
its next higher number. If the left extremities (the numbers
preceding the last two digits) are greater than the product, the right
digit would be the greater option (9,8,7,6) and if left extremities are less
than the product, the right digit would be the smaller option (1,2,3,4).
Let us illustrate the trick with some examples:
Extracting
square root of 784 (√784)
1. Look at the
magnitude of the “hundreds number” (the numbers preceding the last two digits)
which is 7. Now, 22=4 and 32=9. So, the highest
square in 7 is 2 which is the 1stpart of the answer.
2. Now, look at the last
digit of the number which is 4. We know if the number ends in a 4 then the ending digit of the
answer would be 2 or 8.
Now, 2 (findings in step-1) times its next higher
number which is 3 is (2×3=) 6. The left extremity which is 7 is greater than 6.
Therefore, the right digit of the answer must be the greater option which is
8.
So, our final answer is 28.
Let’s go for another example: √3969 (square root of 3969)
1. The magnitude of the
“hundreds number” is 39. Now, 62=36 and 72=49. So,
the highest square in 39 is 6.
2. Looking at the last digit
of the number which is 9; we know if the number ends in a 9 then the last digit
of the answer would be 3 or 7.
Now, 6 (findings in step-1) times its next higher
number 7 is (6×7=) 42. And 39 (the left extremities) is less than 42.
Therefore, the right digit of the answer must be the smaller option i.e.
3.
So, our final answer is 63.
So, square
root of 5476 (√5476) =?
1. The numbers preceding
the last two digits is 54; the highest square in it is 7.
2. The last digit of the
number is 6 so; the ending digit of the answer would be 4 or 6.
Now, 7 times its next higher number (8) is 56.
Since 54 is less than 56, the right digit of the answer must be the smaller
option i.e. 4.
So, our final answer is 74.
Square
root of 13689 (√13689) =?
1. Focusing 136;
the highest square in it is 11 (since, 112 = 121 and 122 =
144).
2. The last digit of the
number is 9 so; the ending digit of the answer would be 3 or 7.
11 times its next higher number (12) is 132 and 136
is greater than 132, so the right digit of the answer would be 7.
So, the final answer is 117.
Square
root of 15376 (√15376) =?
1. The highest square in
153 is 12 (122 = 144 and 132 = 169).
2. The last digit of the
number 6 makes the ending digit of the answer a possibility of 4 or 6.
12 times its next higher number (13) is 156. Since
153 is less than 156, the right digit of the answer must be 4 giving the final
answer 124.
