Problem Set 1 (Surds and Indices) - Quantitative Aptitude Video Lecture - CSAT Preparation - UPSC

let us solve various questions on Anna
troops
and indices which use these laws so the
first question is what is the five by
fourth power of 256 well if you recall
256 is the square of 16 so you can write
256 raise to 5 by 4 as 16 square raise
to 5 by 4 well 16 is the square of 4 so
you can write this as 4 square raised to
2 and which is raised to 5 by 4 now you
know the law of indices if you have a
raise to M raise to n it is equals to a
raise to M times n so this is equals to
4 raised to 2 times 2 is 4 which is
again a raise to 5 by 4 and this is
equivalent to 4 raise to 4 times 5 by 4
this 4 and 4 gets canceled out what
remains is for raise to 5 now 4 raise to
5 is equal to 1 0 to 4 but let us
simplify it more 4 raise to 5 can be
written as 2 raise to 2 raise to 5 4 is
2 square and that is equals to 2 raise
to 10 and most people remember the
powers of 2 so 10 its power of 2 is 1024
moving ahead to the next question what
could be the right answer first of all
let us represent 216 as a product of its
prime factors I hope we can get
something from that
so 216 can be divided by 2 we get 1 0 8
so 2 times 5 is 10 2 times 4 is 8 2
times 27 is 54 then now we cannot divide
by 2 we will divide it by 3 so 3 times 9
is 27 and 3 times
is 9 so 216 is equal to 2 cubed 1 2 3 2
times 3 cube now this is equivalent to 2
times 3 base 2 3 this is the law of
indices if you have a raise to M times B
raised to the same number you can write
it as a B raised to M so that's what we
have done here so this is equivalent to
6 cube so 216 is the cube of 6 so I can
write this as minus 1 upon 6 Q raise to
minus 2 by 3 now we can write minus 1 as
minus 1 cube because minus 1 cube will
give you minus 1 so this is minus 1 cube
upon 6 cube raised to minus 2 by 3 we
can write this as minus 1 by 6 whole
cube then again whole raise to minus 2
by 3 so this is equal to we know the law
of indices which is a raise to M raise
to n is equals to a raise to M times n
so we can write this as minus 1 by 6
raised to minus 2 by 3 times 3 3 & 3
gets cancelled out what remains is minus
1 by 6 raised to minus 2 now if you have
a raise to minus 1 it is equivalent to 1
by a so now we can write this minus 1 by
6 raised to minus 2 as 1 upon minus 1 by
6 raised to 2 and that is equals to 1
upon 1 by 36 and this 36 will go up so
what remains is 36 so the correct answer
is option a 36 okay what could be the
answer of this next question you know
the formula
if a is raised to minus one it is
equivalent to 1 by a so if you have 1 by
a raise to minus 1 it is equivalent to a
so you can write this complete thing as
216 raised to 2 by 3 plus 256 raised to
3 by 4 plus 32 raised to 1 by 5 so you
have it right there and this is
equivalent to 216 is 6 cube so you have
6 cubed raised to 2 by 3 plus 256 is
equivalent to 4 raised to 4
well if you calculate 4 raised to 4 you
will get 256 so this is 4 raised to 4
then we have raised to 3 by 4 plus 32 is
2 raised to 5 the 2 raised to 5 raised
to 1 by 5 and this is equivalent to 6
raised to 2 by 3 times 3 plus 4 raised
to 4 times 3 by 4 plus 2 raised to 5
times 1 by 5 the law of indices that
I've used here is a raise to M raise to
n is equals to a raise to M times n so
this is equivalent to 3 and 3 get
cancelled out here here 4 and 4 get
cancelled out and here 5 & 5 get
canceled up so what remains is 6 square
plus 4 cubed plus 2 raised to 1 so 6
squared is 36 plus 4 cubed is 64 plus 2
and this is equal to 102 so our answer
is option A well this one's an easy
question what would be the answer of
this question so you have a lot of
indices that says that if you have a
raised to M divided by the same number
raised to another power
it is equivalent to a raise to M minus n
so you can write this expression as 10
raised to 150 divided by 10 raised to
146 so this is equivalent to a raise to
M upon a raise to n so it is equal to 10
raised to 150 minus 146 and that is
equals to 10 raised to power 4 and that
is 1 followed by 4 zeros so 10,000 is a
right answer this one is also easy this
is the final question what could be the
answer well you can always use this
formula e raise to M upon ei raised to n
is equal to a raise to M minus n so you
have 1000 raised to 7 upon 10 raised to
18 but you cannot directly use this
formula because this number a should be
same in the numerator and denominator
but we have 1,000 in the numerator and
10 in the denominator well we can
convert this 1000 is 10 cube so this is
equal to 10 raised to 3 raise to 7 upon
10 raised to 18 now there is another
formula a raise to M raised to n is
equals to a raise to M times n so this
is equal to 10 raised to 3 times 7 which
is 21 upon 10 raised to 18 now you can
use this formula and get the answer so
this is equivalent to 10 raised to 21
minus 18 and that is equal to 10 raised
to 3 so 1000 is the right answer