Lab Manual: Algebraic Identity (a³ - b³)=(a-b) (a²+ab+b²) | Lab Manuals for Class 9 PDF Download

Objective

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

• Take 27 cubes and place them to form a stack consisting of a 9 columns, each column consisting 3 cubes [fig. (i)].
• Remove one cube from this stack get a stack of 26 cubes (Arrangement I)

• 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.

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.

• 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
• 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.

Learning Outcome
In this way, students can learn the concept of verifying the identity geometrically by adding different cubes and cuboids.

Activity Time
Students can perform this activity for any other values of a and b e.g. a = 6, b = 2 and find volumes of different cubes and cuboids through this activity.