Calculating Square Root by Prime Factorization - Quantitative Aptitude Video Lecture | Quantitative Aptitude (Quant) - CAT

in this video we will be learning a
method of calculating square roots with
the help of prime factorization and as
always we will do this with the help of
an example let us take the square root
of 2025 we all know that any number can
be expressed as a product of its prime
factors so first of all let us calculate
the product of prime factors of this
number to 0 to 5 so I have 2 0 to 5 I
can start dividing it by the lowest
prime numbers so 2 0 to 5 cannot be
divided by 2 can it be divided by 3 so 5
plus 2 is 7 plus 2 is 9 so the sum is 9
which is divisible by 3 so this number
is divisible by 3 so 3 if it divides to
0 to 5 you get 675 now 675 is also
divisible by 3 let us divide it on
dividing we get 225 again 225 is
divisible by 3 so 3 times 75 is 225 now
75 is again divisible by 3 5 plus 7 is
12 which is divisible by 3 so 3 times 25
is 75 now 25 is not divisible by 3
the next greater prime number is 5 so 5
times 5 is 25 so 2 0 to 5 can be
expressed as a product of prime factors
and these are its prime factors so 3
times 3 times 3 times 3 these are 1 2 3
& 4 3 times 5 times 5 and this is equal
to we can group these numbers in squares
so this is equal to 3 square times 3
square times 5 square now we can take
the under root of both the sides of
equal to now on the right hand side it
is equal to under root of 3 square times
under root of 3 square times and
root of pipe square so the under root
and the square gets cancelled out so
what remains is 3 times 3 times 5 so 3
times 3 is 9 9 times 5 is 45 and under
root of 2 0 to 5 is 45 let us verify
this with the help of a calculator so 45
times 45 should come out as 2025 so 45
times 45 is equal to 2 0 to 5 so our
answer and our method is correct let us
take another example let's calculate the
under root of 250 so let's calculate the
prime factors of 250 first so we have
250 250 is divisible by 2 so 2 times 125
is 250 now 125 is not divisible by 3 it
is divisible by 5 so 5 times 25 is 125
and 25 is 5 times 5 so the prime factors
of 250 are 250 is equal to it can be
written as a product of prime factors
that is 2 times 5 times 5 times 5 now we
can group this in squares so 5 and 5 can
be grouped and 2 and 5 cannot be grouped
so this is equal to 2 times 5 times 5
square right or this can be clubbed to
form 10 5 to the 10 times 5 square if we
take the under root on both the sides of
equal to we get under root of 10 times 5
square which is equal to 5 comes out of
the root under root 10 so this is the
final answer this cannot be solved
further so I have a question for you now
you have to find out the square root of
984 so let us first calculate the prime
factors of 984 so I have 984 it is
divisible by 2 so to tie
4:9 - is 984 this again is divisible by
2 so 2 times 2 4 6 s 4 92 again 2 times
1 2 3 is 2 36 3 plus 2 is 5 plus 1 is 6
6 is divisible by 3 so we can divide it
by 3 so 3 times 4 is 12
bonza 3 3 times 41 is 123 and 41 is a
prime number so you cannot go further so
984 can be written as a product of its
prime factor so they are 2 times 2 times
2 times 3 times 41 now this can be
grouped as 2 square this is as it is 2 3
and 41 so it is equal to 2 square times
2 times 3 times 41 now we can take the
under root on both the sides of equal to
we have under root of 984 is equal to
under root of 2 square times under root
of 2 times under up three times under
root of 41 this is equal to taking the
under root of 2 square we get 2 then we
have root 2 we cannot calculate it right
now then root 3 then route 41 of course
you can calculate the under root of non
perfect square numbers and these come
out in decimal points but right now we
can keep it as it is because we have got
our answer