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)
