The document proves the algebraic identity a3 - b3 = (a - b)(a2 + ab + b2) using unit cubes. It takes values of a = 3 and b = 1. It represents each term of the identity using cubes and cuboids of unit cubes: a3 as a 3x3x3 cube; b3 as a 1x1x1 cube; a3 - b3 as their difference; and the terms of (a - b)(a2 + ab + b2) as cuboids. It is observed that the number of unit cubes in a3 - b3 equals the number in (a - b)(a2 + ab + b2), proving the identity.

1. Algebraic Identities.. By RASHMI KATHURIA

2. Activity 3 Aim : To prove the algebraic identity a 3 -b 3 =(a-b)(a 2 +ab+b 2 ) using unit cubes. Material required: Unit Cubes.

3. Start Working.. Take any suitable value for a and b. Let a=3 and b=1

4. Step 1. To represent (a) 3 make a cube of dimension a x a x a i.e. 3x3x3 cubic units.

5. Step2. To represent a 3 - b 3 extract a cube of dimension b x b x b i.e. 1x1x1 from the cube formed in the step 1 of dimension a x ax a i. e 3x3x3 cubic units. - =

6. Step3. To represent (a-b)a 2 make a cuboid of dimension (a-b) x a x a i.e. 2x3x3 cubic units.

7. Step4 . To represent (a-b) a b make a cuboid of dimension (a-b) x a x b i.e. 2x3x1 cubic units.

8. Step5. To represent (a-b)b 2 make a cuboid of dimension (a-b) x b x b i.e. 2x1x1 cubic units.

9. Step6. To represent (a-b)a 2 +(a-b)ab+(a-b)b 2 I .e (a-b)(a 2 +ab+b 2 ) , join all the cuboids formed in the Steps 3 ,4 and 5. + + =

10. Observe the following The number of unit cubes in a 3 = …27….. The number of unit cubes in b 3 = …1….. The number of unit cubes in a 3 –b 3 = …26….. The number of unit cubes in (a-b)a 2 =…18… The number of unit cubes in (a-b )a b=…6…… The number of unit cubes in (a-b)b 2 =…2…… The number of unit cubes in (a-b)a 2 + (a- b) a b + (a-b)b 2 = …26……

11. It is observed that the number of unit cubes in a 3 -b 3 is equal to the number of unit cubes in (a-b)a 2 +(a-b)ab+(a-b)b 2 i.e. (a-b)(a 2 +ab+b 2 ). Learning Outcome