**NCERT Class 9 Maths Lab Manual – Verify the Algebraic Identity (a – b)³ = a³ – b³ – 3ab (a – b)**

**OBJECTIVE**

To verify the algebraic identity (a – b)³ = a³ – b³ – 3ab (a – b).

**Materials Required**

- Geometry box
- Acrylic sheet
- Scissors
- Adhesive/Adhesive tape
- Cutter

**Prerequisite 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**

- By using acrylic sheet and adhesive tape/adhesive, make a cube of side (a – b) units, where a > b. (see Fig. 8.1)

- By using acrylic sheet and adhesive tape, make three cuboids each of dimensions, (a-b) x a x b. (see Fig. 8.2)

- By using acrylic sheet and adhesive tape make a cube of side b units, (see Fig. 8.3)

- Arrange all the cubes and cuboids as shown in Fig 8.4.

**Demonstration**

In Fig. 8.1, volume of the cube of side (a – b) units = (a – b)³

In Fig. 8.2, volume of a cuboid of sides (a – b) x a x b =(a – b)ab

In Fig. 8.2, volume of three cuboids = 3 x (a – b) ab In Fig. 8.3, volume of the cube of side b = b³

In Fig. 8.4, volume of the solid = Sum of volume of all cubes and cuboids

= (a- b)³ + (a – b) . ab + (a – b) . ab + (a – b) ab + b³

= (a – b)³+3 (a – b) . ab + b³ …(i)

Also, the obtained solid in Fig. 8.4 is a cube of side a.

Therefore, its volume = a³

From Eqs. (i) and (ii), we get

(a – b)³ + 3ab (a – b) + b³= a³

=> (a – b)³ = a³ – b³ – 3ab (a – b)

Here, volume is in cubic units.

**Observation**

By actual measurement,

a = …….. , b = …….. , a-b =…….. ,

So, a³ =…….. , ab = …….. ,

b³ =…….. , ab(a – b) = …….. ,

3ab(a – b) = …….. , (a – b)³ = ……..

Therefore, we observe that

(a-b)³ = a³ – b³ -3ab (a-b) or (a-b)³ =a³-3a²b + 3ab²-b³

**Result**

From above observation, algebraic identity for any a, to, where (a > b) is (a – b)³ = a³ – b³ – 3ab(a – b)

Has been verified geometrically by using cubes and cuboids.

**Application**

This identity is useful in

- many operations of algebraic expressions like as simplification and factorization.
- calculating cube of a number represented as the difference of two convenient numbers.

**Viva Voce**

**Question 1:**

What is the formula of the volume of a cube?

**Answer:**

Volume of a cube = side x side x side = ( side)³

**Question 2:**

What is the formula of the volume of a cuboid?

**Answer:**

Volume of a cuboid = length x breadth x height

**Question 3:**

How would you expand a³ – b³, in the terms of (a – b)³?

**Answer:**

We know that (a – b)³ = a³ – b³ – 3ab(a – b)

= a³ – b³ – 3a²b + 3ab² => a³ – b³ =(a – b)³ + 3a²b – 3ab²

**Question 4:**

What is the expanded form of (a – b)³?

**Answer:**

Expanded form of (a – b)³ = a³ -b³ – 3ab (a – b)

**Question 5:**

Does the resulted value of the product of (a – b)² and (a – b) is same as (a – b)³ ? Give reason.

**Answer:**

Yes, because (a – b)² (a – b) = (a – b)³ = a³ – b³ – 3ab (a – b) [∴ A^{m} x A^{n} = (A)^{m+n}]

**Suggested Activity**

Verify that (x – y)³ =x³ – 3x²y + 3xy² – y³ by taking x = 100 and y = 2.

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