Sunday, July 16, 2023

Class 09 To verify the algebraic identity :a3 – b3 = (a – b)(a2 + ab + b2)

 

Activity 10




OBJECTIVE











To verify the algebraic identity :a3 b3 = (a b)(a2 + ab + b2)

 METHOD OF CONSTRUCTION

MATERIAL REQUIRED

Acrylic sheet, sketch pen, glazed papers, scissors, adhesive, cello-tape, coloured papers, cutter.

1.   Make a cuboid of dimensions (a–b) × a × a (b < a), using acrylic sheet and cellotape/adhesive as shown in Fig. 1.

 2.   Make another cuboid of dimensions (a–b) × a × b, using acrylic sheet and cellotape/adhesive as shown in Fig. 2.

 3.   Make one more cuboid of dimensions (a–b) × b × b as shown in Fig. 3.

 4.    Make a cube of dimensions b × b × b using acrylic sheet as shown in Fig. 4.

5.   Arrange the cubes and cuboids made above in Steps (1), (2), (3) and (4) to obtain a solid as shown in Fig. 5, which is a cube of volume a3 cubic units.

Fig. 5

 Fig. 6

 DEMONSTRATION

 Volume of cuboid in Fig. 1 = (a–b) × a × a cubic units.

 Volume of cuboid in Fig. 2 = (a–b) × a × b cubic units.

 Volume of cuboid in Fig. 3 = (a–b) × b × b cubic units.

 Volume of cube in Fig. 4 = b3 cubic units.

 Volume of solid in Fig. 5 = a3 cubic units.

 Removing a cube of size b3 cubic units from the solid in Fig. 5, we obtain a solid as shown in Fig. 6.

 Volume of solid in Fig. 6 = (a–b) a2 + (a–b) ab + (a–b) b2

 =  (a–b) (a2 + ab + b2)

 Therefore, a3 b3 = (a b)(a2 + ab + b2)

OBSERVATION

 On actual measurement:

 a = ..............,        b = ..............,

 So, a3 = ..............,       b3 = .............., (ab) = ..............,    ab = ..............,

 a2  = ..............,      b2 = ..............,

 Therefore, a3 – b3 = (a – b) (a2 + ab + b2).

 APPLICATION

 The identity may be used in simplification/factorisation of algebraic expressions.

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