1.3 Cubes and Cube roots:
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In the
previous classes we had studied an object called cuboid
which has three dimensions
namely length, breadth and height. Cube is
special figure whose length=breadth=height Size(volume)
of a cube = length*length*length = (length)3 if a =
1,2,3 units, then the volume of cube
respectively is = 1*1*1 = 1,
2*2*2=8, 3*3*3=27 units |
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Is there a
relationship between the numbers 1, 8, 27, 64… and 1, 2,
3, 4… ?
They are
numbers obtained by multiplying the same number thrice from the 2nd
set of numbers.
Cube
numbers are integers raised to the power of 3. They are of the form n3.
If n3
is the cube of number n then n is the ‘cube
root’ of the number n3 .Cube root is denoted by
So 1,2,3,4
are cube roots of the numbers 1, 8, 27, 64 respectively.
Definition:
Any number which is a product of three identical numbers is called ‘cube’.
The volume
of a solid figure of three dimension=length*breadth*depth(we
learn this in Geometry).
Cube is a
solid figure whose length, breadth and depths are same.
The volume of a cube =
length*length*length = (length)3
1.3 Problem 1 :
If the side of a cube is 7cms find its volume
Solution:
Volume of
a cube = (length)3
Volume of cube of side 7cms length= (7)3
=7*7*7 =
343 cubic cms
1.3.1 Finding cube root by factorisation
In this
method we find all prime factors of the given number and then group the common
factors in triples, such that all two factors in each pair are the same. If
some factors do not appear in triples then the number is not a perfect cube and
we stop the process of grouping.
1.3.1 Problem 1 :
Find the cube root of 42875
Solution:
The
factors of 42875 are 5,5,5,7,7,7
42875 = 5*5*5*7*7*7 = 53*33=
(5*7)3
= 5*7 =35
1.3.1 Problem 2: Find
the least number by which 432 must be multiplied or divided to make it a
perfect cube.
Solution:
The
factors of 432 are 3,3,3,4.4 = (3)3*(4)2
We note
that the factor 4 appears only 2 times.
1. If we
multiply 432 by 4 we get 432*4 = (3)3*(4)3= (3*4)3,
so when the smallest number 4 multiplies 432, it gives us the cube number
(=1728=123)
2. If we
divide 432 by 16 we get 432/16= 33, so when the smallest number 16
divides 432, it gives us the cube number (=27 = 33).
1.3.1 Problem 3: If
1 cubic cc of water weighs 1 gram, what is the weight of 1 cubic meter of
water?
Solution:
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Volume
of cube of side of 1meter (Note: 1m=100cm) = side*side*side= (side)3cc
= (100)3cc = 1000000cc Since 1
cc of water weighs one gram 1000000cc
of water will weigh 1000000gm = 1000 kg |
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1.3 Summary of
learning
|
No |
Points studied |
|
1 |
Cubes , finding cube root by factorisation method |
Additional Points:
Properties
of perfect cubes and cube roots:
1. The
relationship between unit’s digit of cube and its cube root is as follows:
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Unit’s digits of cube ==è |
0,1,4,5,6,9 |
2 |
8 |
3 |
7 |
|
Unit’s digit of cube root ==è |
Same as that of unit’s digits of cube |
8 |
2 |
7 |
3 |
2. If a number is negative, its cube is also
negative ((-7)3 = -343: 
= -7)
3. Cube of an even number is even and cube of an
odd number is odd
4. Cube root of a rational number = cube root of
numerator/cube root of denominator (
= 
/
= 7/5)