Using unit cubes 4

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This is an activity to verify the formula (a-b)^3

This is an activity to verify the formula (a-b)^3

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  • 1. Algebraic identities… 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-b) 3 make a cube of dimension (a-b) x (a-b) x (a-b) i.e. 2x2x2 cubic units.
  • 5. Step 2. To represent (a) 3 make a cube of dimension a x a x a i.e. 3x3x3 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 a 3 + 3ab 2 , join the cube and the cuboids formed in steps 2 and 3. + =
  • 8. Step 5. To represent a 3 + 3ab 2 - 3a 2 b extract from the shape formed in the previous step 3 cuboids of dimension 3x3x1. - =
  • 9. Step 6. To represent a 3 + 3ab 2 - 3a 2 b-b 3 extract from the shape formed in the previous step 1 cube of dimension 1x1x1. - =
  • 10. Step 7. Arrange the unit cubes left to make a cube of dimension 2x2x2 cubic units.
  • 11. Observe the following
    • The number of unit cubes in a 3 = …27…..
    • The number of unit cubes in 3ab 2 =…9……
    • The number of unit cubes in 3a 2 b=…27……
    • The number of unit cubes in b 3 =…1……
    • The number of unit cubes in
    • a 3 - 3a 2 b + 3ab 2 - b 3 = …8…..
    • The number of unit cubes in (a-b) 3 =…8…
  • 12. 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