The document presents an algebraic identity proof using unit cubes. It takes values of a=3 and b=1, and represents: a3 as a 3x3x3 cube. 3a2b as 3 cuboids of dimension 3x3x1. 3ab2 as 3 cuboids of dimension 3x1x1. b3 as a 1x1x1 cube. Joining these representations results in a 4x4x4 cube, demonstrating the identity (a+b)3 = a3 +3a2b+3ab2 +b3.

1. Algebraic Identities A presentation by.. Rashmi Kathuria

2. Activity Aim : To prove the algebraic identity (a+b) 3 = a 3 +3a 2 b+3ab 2 +b 3 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. Step 2. To represent 3a 2 b make 3 cuboids of dimension a x a x b i.e. 3x3x1 cubic units.

6. Step 3. To represent 3ab 2 make 3 cuboids of dimension a x b x b i.e. 3x1x1 cubic units.

7. Step 4. To represent b 3 make a cube of dimension a x a x a i.e. 1x1x1 cubic units.

8. Step 5. Join all the cubes and cuboids formed in the previous steps to make a cube of dimension (a +b) x ( a +b) x ( a +b) i.e. 4x4x4 cubic units. = + + +

9. Observe the following The number of unit cubes in a 3 = ..27….. The number of unit cubes in 3a 2 b =…27… The number of unit cubes in 3ab 2 =…9…… The number of unit cubes in b 3 =…1…… The number of unit cubes in a 3 + 3a 2 b + 3ab 2 + b 3 = ..64…. The number of unit cubes in (a+b) 3 =…64…

10. It is observed that the number of unit cubes in (a+b) 3 is equal to the number of unit cubes in a 3 +3a 2 b+3ab 2 +b 3 . Learning outcome