**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**

- Acrylic sheets
- Geometry box
- Adhesive/Cello-tape
- Cutter
- Scissors

**Prereequisite Knowledge**

- Concept of cuboid and its volume.
- Concept of cube and its volume.

**Theory**

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

**Procedure**

- 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)

- Make a cuboid of dimensions a x a x b using acrylic sheet and cello-tape/adhesive, (see Fig. 9.3)

- Make a cuboid of dimensions a x b x b using acrylic sheet and cello-tape/adhesive, (see Fig. 9.4)

- Arrange these cubes and cuboids as shown in Fig. 9.5.

- 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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