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

 Activity 9

 OBJECTIVE     

                                                             

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

  MATERIAL REQUIRED

Acrylic sheet, glazed papers, saw, adhesive, cellotape, coloured papers, sketch pen, etc.

 METHOD OF CONSTRUCTION

 1.   Make a cube of side a units and another cube of side b units as shown in Fig. 1 and Fig. 2 by using acrylic sheet and cellotape/adhesive.

 2.   Make a cuboid of dimensions a × a × b [see Fig. 3].

 3.   Make a cuboid of dimensions a × b × b [see Fig. 4].

 Arrange these cubes and cuboids as shown in Fig.

DEMONSTRATION

 Volume of cube in Fig. 1 = a³

 Volume of cube in Fig. 2 = b³

 Volume of cuboid in Fig. 3 = a²b

 Volume of cuboid in Fig. 4 = ab²

 Volume of solid in Fig. 5 = a³+b³ + a²b + ab² = (a+b) (a² + b²)

 Removing cuboids of volumes a²b and ab², i.e.,Fig. 6

ab (a + b) from solid obtained in Fig. 5, we get the solid in Fig. 6.

 Volume of solid in Fig. 6 = a³ + b³.

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

 = (a+b) (a² + b² – ab)

 Here, volumes are in cubic units.

 OBSERVATION

 On actual measurement:

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

 So, a³ = ..............,       b³  = .............., (a+b) = ..............,    (a+b)a² = ..............,

 (a+b) b²  = ..............,       a²b = ..............,             ab² = ..............,

 ab (a+b)  = ..............,

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

 APPLICATION

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