Algebraic identities… By Rashmi Kathuria…
Activity <ul><li>Aim :  To prove the algebraic identity </li></ul><ul><li>(a-b) 3  = a 3  -3a 2 b+3ab 2 -b 3  </li></ul><u...
Start Working.. Take any suitable value for  a  and b. Let a=3  and b=1
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.
Step 2.  To represent  (a) 3   make a cube of dimension  a x a x a  i.e. 3x3x3  cubic units.
Step 3.  To represent  3ab 2  make 3 cuboids  of dimension  a x b x b i.e. 3x1x1 cubic units.
Step 4.  To represent  a 3  + 3ab 2  ,  join the cube and the cuboids formed in steps 2 and 3.   + =
Step 5.  To represent  a 3  + 3ab 2 - 3a 2 b  extract from the shape formed in the previous step 3 cuboids of dimension 3x...
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 ...
Step 7.  Arrange the unit cubes left to make a cube of dimension 2x2x2 cubic units.
Observe the following <ul><li>The number of unit cubes in a 3  = …27….. </li></ul><ul><li>The number of unit cubes in 3ab ...
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 ...
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Using unit cubes 4

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

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Using unit cubes 4

  1. 1. Algebraic identities… By Rashmi Kathuria…
  2. 2. Activity <ul><li>Aim : To prove the algebraic identity </li></ul><ul><li>(a-b) 3 = a 3 -3a 2 b+3ab 2 -b 3 </li></ul><ul><li>using unit cubes. </li></ul><ul><li>Material required : Unit Cubes. </li></ul>
  3. 3. Start Working.. Take any suitable value for a and b. Let a=3 and b=1
  4. 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. 5. Step 2. To represent (a) 3 make a cube of dimension a x a x a i.e. 3x3x3 cubic units.
  6. 6. Step 3. To represent 3ab 2 make 3 cuboids of dimension a x b x b i.e. 3x1x1 cubic units.
  7. 7. Step 4. To represent a 3 + 3ab 2 , join the cube and the cuboids formed in steps 2 and 3. + =
  8. 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. 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. 10. Step 7. Arrange the unit cubes left to make a cube of dimension 2x2x2 cubic units.
  11. 11. Observe the following <ul><li>The number of unit cubes in a 3 = …27….. </li></ul><ul><li>The number of unit cubes in 3ab 2 =…9…… </li></ul><ul><li>The number of unit cubes in 3a 2 b=…27…… </li></ul><ul><li>The number of unit cubes in b 3 =…1…… </li></ul><ul><li>The number of unit cubes in </li></ul><ul><li>a 3 - 3a 2 b + 3ab 2 - b 3 = …8….. </li></ul><ul><li>The number of unit cubes in (a-b) 3 =…8… </li></ul>
  12. 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

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