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Sunday, December 24, 2006

Verifying Algebraic identity(II)

Today I am sharing with U the 2nd cubic algebraic activity which can be done
using the unit cubes.
•To prove the algebraic identity
(a-b)^3 = a^3 -3a^2b+3ab^2-b^3
using unit cubes.
Take any suitable value for a and b.
Let a=3 and b=1.
To represent (a-b)^3 make a cube of dimension
(a-b) x (a-b) x (a-b) i.e. 2x2x2 cubic units as shown below.

To represent (a)^3 make a cube of dimension a x a x a
i.e. 3x3x3 cubic units as shown below.

To represent 3ab^2 make 3 cuboids of dimension
a x b x b i.e. 3x1x1 cubic units as shown below.

To represent a^3 + 3ab^2 , join the cube and the cuboids
formed in steps 2 and 3 as shown below.

To represent a^3 + 3ab^2- 3a^2b extract from the shape
formed in the previous step 3 cuboids of dimension
3x3x1 to get the shape shown below.
To represent a^3 + 3ab^2- 3a^2b-b^3 extract from the
shape formed in the previous step 1 cube of dimension
1x1x1 .The shape shown below will be obtained.

Arrange the unit cubes left to make a cube of dimension
2x2x2 cubic units.

Observe the following
•The number of unit cubes in a^3 = …27…..
•The number of unit cubes in 3ab^2 =…9……
•The number of unit cubes in 3a^2b=…27……
•The number of unit cubes in b^3 =…1……
•The number of unit cubes in
a^3 - 3a^2b + 3ab^2- b^3 = …8…..
•The number of unit cubes in (a-b)^3 =…8…
It is observed that the number of unit cubes
in (a-b)^3 is equal to the number of unit cubes
in a^3 -3a^2b+3ab^2-b^3 .
Using the same strategy we can verify the other cubic identities.

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