Calculation Tricks : Squares and Cubes

Calculation Tricks will be a series of articles to aid you in handling all kinds of calculations that can be done faster. The phase 1 of this learning involves understanding the method while phase 2 involves practicing it on basic problems on the concepts. Eventually, phase 3 on your end will involve imbibing the methods into all your calculations as the methods discussed will be specific to situations so taking out the right weapon at the right occasion is an art you will have to hone by practice.


The first article today will be focused upon calculating squares and cubes of any number with a detailed focus on squares as cubes are not a frequent occurrence in calculations.
Prerequisite : Must know squares from 1-25(actually till 30 is advisable) as presented in the image below for quick reference. Do not use calculators,pen or paper while going through these articles or while practicing these concepts as it would defeat the whole purpose. You need to master this series with only mental means and completely take the physical means out of running.




Concept 1 : Squares of numbers ending with the digit 5

Let the number be written as n5. Then the square is given as : n(n + 1) | 25 where | indicates just a separator to visualize the squared number in two parts.

examples : 15^2 = 1*(1 + 1) | 25 = 225 , 45^2 = 4*5 | 25 = 2025 , 135^2 = 13*14 | 25 = 18225

Practice yourself : 35^2 , 225^2 , 1005^2

Concept 2 : Finding squares 26 - 75

The numbers in this range can be seen as (50 - x) for 26-49 or (50 + x) 51 - 75.
The square in this scenario is given as : (25 -+ x) | x^2 where the right hand part(x^2) must be written as a two digit number only. Hence 3^2 = 9 will be written as 09 while in case of a three digit square for right hand part, the left most digit of the square is added as carry to left hand part.

examples : 44^2 = (25 - 6) | 6^2 = 1936 , 53^2 = (25 + 3) | 3^2 = 2809 , and
64^2 = (25 + 14) | 196 = (39 + 1) | 96 = 4096
Practice yourself : 37^2 , 62^2, 73^2


Concept 3 : Finding squares 75 - 130

The numbers in this range can be seen as (100 - x) or (100 + x). Right hand part must again be two digit number only while left hand part is calculated as (100 +- 2x).

examples : 93^2 = (100 - 2*7) | 7^2 = 8649 , 113^2 = (100 + 26) | 13^2 = (126 + 1) | 69 = 12769

Pick examples for yourself in this range and practice.


Concept 4 : Finding squares 175 - 230 , 275 - 330 and so on..


The concept primarily remains same as concept 3 with the only change in base of our square calculation. These are calculated : (200 +- 2x)*2 | x^2 where the right hand part again remains as a two digit number since we are dealing in hundreds.

examples : 202^2 = (200 + 2*2)*2 | 04 = 40804 , 193^2 = (200 - 2*7)*2 | 49 = 37249

Practice your own examples to get familiar with the concept.

For 275 - 300 range we take (300 +- 2x)*3 | x^2 and so on for 875 - 930 range..


Concept 5 : Find squares of n digit numbers where n >= 4 i.e thousands and beyond

The change for 1000s , 10000s and beyond is that right hand part is taken as a three digit number, four digit number and so on respectively..
examples :
1017^2 = (1000 + 2*17) | 17^2 = 1034 | 289 = 1034289 [ No carries this time ]
1994^2 = (2000 - 2*6)*2 | 6^2 = 3976 | 036 = 3976036  and so on..


Before moving onto cubes, did anyone think why are we able to do this ?
(50 +- x)^2 = 2500 + x^2 +- 100x = (25 +- x)*100 + x^2 = (25 + x)|x^2
(100 + -x)^2 = 10000 + x^2 +- 200x = (100 +- 2x)*100 + x^2 = (100 +-2x)|x^2
and  so on..
Whenever you learn shortcuts, do try to be curious about the concept behind it. Would give you a more generic outlook to the shortcut..


Concept 6 : Finding cubes

We ll discuss the method for finding two digit cubes as more than that is not useful for our purpose nor necessary. The prerequisite for this shortcut is you knowing the first 10 cubes by heart.
1^3 = 1 , 2^3 = 8 , 3^3 = 27 , 4^3 = 64 , 5^3 = 125 , 6^3 = 216 , 7^3 = 343 , 8^3 = 512 , 9^3 = 729 and 10^3 = 1000 [ Advisable to learn up to 15 if you have the time ]


Let's learn the calculation technique then :




That's about it you will ever need for calculating squares and cubes. Next article in this series would take up some other aspect of calculation part involved in any of your MBA exam papers. If you like the content here on Lagom, please help spread the word by sharing,liking and commenting on posts about the content.


Until next time..
Cheers !
AS