Transcript cubes and cube roots_Chapter7

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 Introduction
 Perfect
Cubes
 Cube root
 Cube root by prime
factorisation
 Cube root by estimation.

one of the great mathematical geniuses, S
ramanujan had a visit of prof G H hardy. He
came with a taxi whose number is 1729 .he
described the number as a dull number.
Ramanujan quickly pointed that 1729 was
indeed interesting. He said it was the
smallest number that can be expressed as
the sum of two cubes in two different ways.
1729 =
3
1728+1=12
+
3
1
1729 = 1000+729=103 + 93
Numbers like 1, 8 ,27…
are
called
cube
numbers
or
a
perfect cube . We get
perfect cubes by
multiplying a number
3 times with the same
number.
A
3

For example- 1×1×1 = 13 or 1cube, 8= 23 ,
No.
Cube.
27= 33…etc.
1
1
2
8
3
27
4
64
5
125
6
216
7
343
8
512
9
729
10
1000
11
1331
12
1728
13
2197
14
2744
15
3375
16
4096
17
4913
18
5332
19
6859
20
8000
CUBE ROOTS



Finding the square root, as you know , is the
inverse operation of squaring. Similarly finding
the cube root is the inverse operation of finding
cube.
We know that 23= 8; so we say that cube root of 8
is 2. We write 3 8 = 2.
83=512;so of
3
THE SYMBOL “
”
512 =8.
3
” DENOTES “CUBE ROOT
CUBE ROOT THROUGH FACTORISATION
METHOD
CONSIDER “3375”:
3375=3×3×3×5×5×5
‾‾‾‾‾ ‾‾‾‾‾
=33 × 53=(3 × 5)3
3
3375
= 15
We find its cube root by prime factorisation.
The factors are ;
Therefore, cube root of 3375= 15

EXAMPLE ;
Find the cube root of 8000.
ANSWER:
Prime factorisation of 8000 is
8000=2×2×2×2×2×2×5×5×5
Therefore “ 3 8000 ” = 2×2×5=20
CUBE ROOT BY ESTIMATION
METHOD
To find the cube root of a cube number,
the following method can be used.

STEP 1.
857375= 857
↓
375
↓
second
number
number
Take
a cube
number;first
857375.
We getMake
375 &
857
as
two
groups
of
group of three digits
three digits each
starting from the right most digit
of the number.

STEP 2
375
‾
So,we get 5 at the unit’s place cube root.
First
group
i.e.,375
will
give
the
one’s
 STEP 3
digit of the857
required cube root.
We
that375
9^3=729
10^3=1000.Also,
Theknow
number
ends&with
5.We know
729<857<1000.We take the one”s place,place the
that 5 comes at the unit’s place of a
as the ten “s place of the required cube root.So,we
get
“CR”857375=95
number
only when it’s cube root ends in 5.
Now we take the next group
WORK SHEET (FOR FA-3)
Find the cube root of each of the of the following by prime factorization method.
 64
 512
Find the cube root through estimation
 i.17576
 ii. 3375
 iii.1331
ANSWER






i. 3√64=2×2×2×2×2×2=2×2=4
‾‾‾‾‾ ‾‾‾‾‾
ii.3√512=2×2×2×2×2×2×2×2×2 =2×2×2=8
‾‾‾‾‾ ‾‾‾‾‾ ‾‾‾‾‾
iii.3√10648=11×11×11×2×2×2=11×2=22
‾‾‾‾‾‾‾ ‾‾‾‾‾
i. 26
ii 15
iii 11
WHICH OF THE FOLLOWING ARE NOT PERFECT
CUBE
1. 216
2. 128
3. 1000

ANS – 128 IS NOT A PERFECT CUBE
1.
IS 68600 A PERFECT CUBE ? IF NOT FIND THE
SMALLEST NUMBER BY WHICH IT SHOULD BE
MULTIPLIED TO GET A PERFECT CUBE.
ANS – NO It’S NOt A PeRFect cUBe. It SHOULD Be
MULTIPLIED BY 5
 IS
1188 A PERFECT CUBE ?IF NOT, BY
WHICH SMALLEST NATURAL NUMBER
SHOULD IT BE DIVIDED SO THAT THE
QUOTIENT IS A PERFECT CUBE.
ANS –NO It’S NOt A PeRFect cUBe , It
SHOULD BE DIVIDED BY 44 TO GET A
PERFECT CUBE.