Verify the identity a³+b³ = (a+b)(a²-ab+b²) ;
Answers
Step-by-step explanation:
Procedure
Make a cube of side a units and another cube of side b units by using acrylic sheets and cello-tape/adhesive
- Verify the Algebraic Identity a³+b³ = (a+b) (a²-ab+b²) 1
Make a cuboid of dimensions a x a x b using acrylic sheet and cello-tape/adhesive
Make a cuboid of dimensions a x b x b using acrylic sheet and cello-tape/adhesive,
Arrange these cubes and cuboids
On removing cuboids of dimensions a x a x b and a x b x b from the solid obtained to get another solid as shownVerify the Algebraic Identity a³+b³ = (a+b) (a²-ab+b²) 5
Demonstration
volume of cube of side a units = a³
volume of cube of side b units = b³
volume of cuboid of dimensions a, a and b = a²b
volume of cuboid of dimensions a, b and b = ab²
Volume of the solid obtained in Fig. 9.5 = Total volume of all cubes and cuboids
= a³ + b³ +ab² +a²b = a²(a + b) + b²(a + b)
= (a + b) (a² + b²)
After removing cuboids of volume a²b (i.e. a x a x b) and ab² (i .e. a x b x b) from solid obtained
So, volume of solid in Fig. 9.6 = (a + b) (a² + b²)- a²b – ab²
= (a + b) (a² + b² )-ab (a + b) = (a+ b) (a² + b² – ab) Also, volume of solid in Fig. 9.6 = a³ + b³ Hence, a³+b³ = (a+b) (a²-ab+b²).
Here, volume is in cubic units.
Observation
On actual measurement, we get
a = ……….. , b = ……….. ,
So, a³ = ……….. , b³ = ……….. , a + b = ……….. ,
(a + b) a² = ……….. , (a + b)b² = ……….. ,
a²b = ……….. , ab² = ……….. ,
ab(a + b) = ……….. ,
Hence, a³+b³ = (a+b) (a²-ab+b²).
Result
The algebraic identity a³+b³ = (a+b) (a²-ab+b²) has been verified.