Question

Verify the identity

${a}^{3} + {b}^{3} = ( {a}^{2} - ab \: + {b}^{2} )$
Please answer in steps please​

Answer 1

Answer:

Algebraic 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.

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)

CBSE Class 9 Maths Lab Manual – Algebraic Identity (a3 – b3) = (a – b) (a2 + ab + b2) 1

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

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).

Answer 2

Step-by-step explanation:

tex]a3 + b3 = (a + b)(a2 - a2 - ab + b2)[/tex]