NCERT Class 9 Maths Lab Manual – Verify the Algebraic Identity a³+b³ = (a+b) (a²-ab+b²)

NCERT Class 9 Maths Lab Manual – Verify the Algebraic Identity a³+b³ = (a+b) (a²-ab+b²)

OBJECTIVE

To verify the algebraic identity a³+b³ = (a+b) (a²-ab+b²).

Materials Required

1. Acrylic sheets
2. Geometry box
3. Adhesive/Cello-tape
4. Cutter
5. Scissors

Prereequisite Knowledge

1. Concept of cuboid and its volume.
2. Concept of cube and its volume.

Theory

1. For concept of cuboid and its volume refer to Activity 7.
2. For concept of cube and its volume refer to Activity 7.

Procedure

1. Make a cube of side a units and another cube of side b units by using acrylic sheets and cello-tape/adhesive, (see Fig. 9.1 and 9.2)
   
2. Make a cuboid of dimensions a x a x b using acrylic sheet and cello-tape/adhesive, (see Fig. 9.3)
   
3. Make a cuboid of dimensions a x b x b using acrylic sheet and cello-tape/adhesive, (see Fig. 9.4)
   
4. Arrange these cubes and cuboids as shown in Fig. 9.5.
   
5. On removing cuboids of dimensions a x a x b and a x b x b from the solid obtained in Fig. 9.5, to get another solid as shown in Fig. 9.6.
   

Demonstration
In Fig. 9.1, volume of cube of side a units = a³
In Fig. 9.2, volume of cube of side b units = b³
In Fig. 9.3, volume of cuboid of dimensions a, a and b = a²b
In Fig. 9.4, 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 in Fig. 9.6.
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.

Application
The identity can be used in simplification and factorization of algebraic expressions.

Viva Voce
Question 1:
What is the expanded form of a³ +b³?
Answer:
Expanded form of a³+b³ = (a+b) (a²-ab+b²).

Question 2:
What is the condition which satisfy the condition a³ + b³ = 0 ?
Answer:
a = -b

Question 3:
For evaluating (90)³ + (10)³, which identity you should follow?
Answer:
a³+b³ = (a+b) (a²-ab+b²) will be appropriate.

Question 4:
How would you express (a+b)(a² -ab+b²) in shortest form?
Answer:
a³+ b³.

Question 5:
What is the degree of an algebraic identity a³ +b³?
Answer:
The degree of given algebraic identity is 3.

Question 6:
Let a and b be the edges of two cubes. Is a³ + b³ equal to the sum of volumes of cubes?
Answer:
Yes

Suggested Activity
By taking a = 3 and b = 7, verify the identity a³ +b³.

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