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pls explain how to do this activity in a simple way



Identity (a3 – b3) = (a – b) (a2 + ab + b2)

Objective
To verify the identity a3 – b3 = (a – b)(a2 + ab + b2) geometrically by using sets of unit cubes.

Prerequisite Knowledge
Volume of a cube = (Edge)3
Volume of a cuboid = l x b x h
a3 – b3 = (a – b)(a2 + ab + b2)

Materials Required
A set of 53 plastic or wooden cubes each of dimensions (1 x 1 x 1 unit)

Procedure
To verify a3 – b3 = (a – b)(a2 + ab + b2). Let a = 3 and b =1.

  1. Take 27 cubes and place them to form a stack consisting of a 9 columns, each column consisting 3 cubes [fig. (i)].
  2. Remove one cube from this stack get a stack of 26 cubes (Arrangement I)
    cbse-class-9-maths-lab-manual-algebraic-identity-a3-b3-a-b-a2-ab-b2-2-1
  3. Make arrangement II of 26 cubes. This arrangement consists of three stacks.
    • The first stack consists of 18 cubes such as 9 columns of two cubes each. .
    • The second stack consists of 6 cubes such as two rows of three cubes each.
    • Third stack consist of 1 row of 2 cubes.
      cbse-class-9-maths-lab-manual-algebraic-identity-a3-b3-a-b-a2-ab-b2-2-2

Observation
Since the two arrangements have equal number of cubes (each arrangement has 26 cubes), the total volume in both the arrangements must be equal.

  1. Volume of arrangement I
    Volume of stack in fig. 1(i) = a3
    Volume of stack in fig. 1(ii) = b3
    ∴Volume of arrangement I = Volume of stack in fig. 1(i) – Volume of stack in fig. 1(ii) = a3 – b3
  2. Volume of arrangement II
    Volume of the stack in fig. 2 (i) = (a – b) a2
    Volume of the stack in fig. 2(ii) = (a – b)ab
    Volume of the stack in fig. 2 (iii) = (a – b)b2
    Total volume of arrangement II = (a – b)a2 + (a – b)ab + (a – b)b2 = (a – b)(a2 + ab + b2).
    Since number of cubes in arrangement I and II are equal.
    ∴a3 – b3 = (a – b)(a2 + ab + b2).

Result
The identity a3 – b3 = (a – b)(a2 + ab + b2) is verified geometrically by using cubes and cuboids

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