Examples: Perfect Cubes Video Lecture - Class 8

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sixth question timeline now let's try
two questions question number one find
the smallest number by which each of the
following numbers must be multiplied to
obtain a perfect cube okay so we are
given two numbers 72 and 675 so let's
start with the first number so what do
we do our first step is to do prime
factorization so 72 is completely
divisible by 2 so 2 into 36 again 2 into
18 two nines at 18 then 3 threes and 9
so 72 can be written as 2 into 2 into 2
into 3 into 3 so let's divide it into
groups of 3 2 appears three times but 3
appears only two times so therefore this
can be written as 2 cubed into 3 into 3
so in order to obtain a perfect you
water that number by which this should
be multiplied if we multiply it by
another 3 then our job is done because
if there is one more 3 then it would
become 3 2 so therefore this number must
be multiplied by 3 so we can say that it
must be multiplied by 3 now what will
happen after it is multiplied by P then
it would become 2 cubed into 3 cubed
that is equal to 2 into 3 cubed which is
equal to 6 cube right so this would
become a focus you similarly next look
at the second one which is 675 so here
also first perform line factorization
which is completely divisible by 5 5 on
the file 5 to the 15 5.2 25 again 5 to
the 10 5 7 2 35 + 3 9 27 3 three's a 9
so 675
be written as 5 into 5 into 3 into 3
into 3 so when we divide into groups of
three is three hours three times but
five occurs only two times so that means
we need to divide is we need to multiply
it by five so if you multiply it by five
then what will then happen so in that
case you will have five is cube into
three cube this is equal to 16 - so this
will be 10 becomes perfect cube maybe
number to find the smallest number by
which each of the following numbers must
be divided to obtain a perfect cube so I
think the approach would remain the same
just that instead of multiplying this
time what is that number by which we
should divide so the numbers are 135 and
192 so let us first do prime
factorization of 135 so 5 into 27 three
nines or twenty thousand three Caesar 9
therefore 135 can be written as 5 into 3
into 3 into 3 divided into groups of 3 3
instance occurs 3 lines would find of
course only one so that means this
number must be divided by 5 because
which you divide it by 5 then 5 5 will
cancel each other to the Spy's will be
gone and we will be left with 3 cubed
which is a positive so this number must
be divided by 5 what about 192 here also
the same technique to 918 - desert well
for the 8 - 8 - 16 - to the 4 - 4 the 8
to 12 to 24 to 12 2006
therefore 192 can be written as 2 into 2
into 2 into 2 into 2 into 2 into 3
groups of 3 2 occurs two three times
again two of those three times but three
is left alone so this number must be
divided by 3 so that we can get rid of
this extra fee so the moment is
extracting 1 we will be left with 2 cube
into - 2 which is equal to 4 2 which
isn't permit to question number 3
but makes a few void of classism of
size five centimeter two centimeter eyes
regulator system how a cuboid looks like
so cuboid has size five centimeters
let's play this is 2 centimeters and
this is 5 centimeter again how many such
you boys really need to form a cube now
how can you need to this a cuboid by
birth placing multiple cuboid like this
such that when you make that arrangement
all the three sides should be equal now
let's think logically how do you think
we can place Q points of this side so
that we get a Q so now when you look at
the cuboid two sides are equal that is
the length and the breadth are equal
both are 5 centimeter but the height is
2 centimeter so that means when we are
trying to place more two boys above
let's see if we play sighs you boys ever
so the height will become 5 into 2 that
is 10 centimeters so in that case and
built and the length should also be 10
centimeters so we will have to place two
cuboids side-by-side so that the length
also becomes 10 centimeters then are you
getting my point
okay let's look at another way of
looking at it first of all let us find
out the volume of the cuboid that is
volume of this cube Boyd so volume of
the cuboid is given by length into
breadth into height which is equal to 5
into 2 into 5 that is equal to 50
centimeter cube this is the volume of
this pure white right now first let's
check if this is P is a perfect cube or
not so is to be a perfect cube no this T
is not a perfect you in fact 50 is not a
perfect cube that is why this is not a
cube that is why it is a cuboid and by
now we understand any number it is a
perfect cube that number will be able to
be the volume of a cube so in this case
this is a cuboid 250 is not a perfect
cube now let us see let us do prime
factorization of 15 why am I doing this
we I'll tell you so we see when you look
at it
9 factorization it is 2 into 5 into 5 we
can again prove that 50 is not a perfect
do that that is why this is the cuboid
is not askew now if there any number by
which we can multiply this number such
that it becomes a particular because we
want many more cuboid to be placed so
that we get a cube so any cube is formed
by all numbers within the perfect so do
you think that we can multiply this part
with any number such that it becomes the
perfect cube here the first because here
you see 5 occurs 2 times to make it a
perfect cube we need one more file right
so one more pair is needed so we need 2
occurs one time so to make it a cube 2
more to join resist
so we need two tools and one side so
basically this number must be multiplied
by 2 tools that is 2 into 2 and 1 fine
that is fine so 2 into 2 into 5 so if
you multiply the given number with this
number then this becomes a perfect you
how so when you multiply this it becomes
2 into 2 into 2 into 5 into 5 into 5
which is equal to 2 cube into 5 cube
that is equal to 2 into 5 cube which is
equal to 10 cube so this 10 cube is a
perfect cube this basically means a cube
with each side is equal to 10
centimeters to send you this 1,000 so
1,000 centimeters cube would be the
volume of that cube that that each side
is equal to 10 centimeter so that means
how many such cuboid do you need to make
this cube of 1000 centimeter volume you
would need these limit avoid 2 into 2
into 5 what is 2 into 2 into 5 this is
20 so you actually need 20 cuboid to
make a cube of volume 1000 centimeter
square such that each side of the cube
is 10 cent
so let us look at the arrangement now
you see if you have these two one so if
you place 22 by it like this so first
let's talk about the height so we have
to make the height attain because we saw
that each side of the cube has to be 10
centimeter so right now each side is 2
this is 2
now when you please find such cuboid one
above another then your total height
becomes how much this total height
becomes 10 then now we also want the
length to be 10 so right now this length
is 5 so let us place another cube oil
here so then 5 plus 5 altogether this
becomes 10 to be please you voice like
this so right now the highest and the
length is 10 what about the width the
width is right now 5 so what do we do
displays a cuboid in front of this so
this is also fine so now the wig has
also become all together 10 so therefore
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