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Verify the Algebraic Identity (a – b)³ = a³ – b³ – 3ab (a – b) Experiment Class 9 Maths Practical NCERT

The experiment to determine Verify the Algebraic Identity (a – b)³ = a³ – b³ – 3ab (a – b) are part of the Class 9 Maths Lab Manual provides practical activities and experiments to help students understand mathematical concepts effectively. It encourages interactive learning by linking theoretical knowledge to real-life applications, making mathematics enjoyable and meaningful.

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

Determine Verify the Algebraic Identity (a – b)³ = a³ – b³ – 3ab (a – b) Class 9 Practical

OBJECTIVE

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

Materials Required

  1. Geometry box
  2. Acrylic sheet
  3. Scissors
  4. Adhesive/Adhesive tape
  5. Cutter

Prerequisite 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. By using acrylic sheet and adhesive tape/adhesive, make a cube of side (a – b) units, where a > b. (see Fig. 8.1)
    NCERT Class 9 Maths Lab Manual - Verify the Algebraic Identity (a - b)³ = a³ - b³ - 3ab (a - b) 1
  2. By using acrylic sheet and adhesive tape, make three cuboids each of dimensions, (a-b) x a x b. (see Fig. 8.2)
    NCERT Class 9 Maths Lab Manual - Verify the Algebraic Identity (a - b)³ = a³ - b³ - 3ab (a - b) 2
  3. By using acrylic sheet and adhesive tape make a cube of side b units, (see Fig. 8.3)
    NCERT Class 9 Maths Lab Manual - Verify the Algebraic Identity (a - b)³ = a³ - b³ - 3ab (a - b) 3
  4. Arrange all the cubes and cuboids as shown in Fig 8.4.
    NCERT Class 9 Maths Lab Manual - Verify the Algebraic Identity (a - b)³ = a³ - b³ - 3ab (a - b) 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

  1. many operations of algebraic expressions like as simplification and factorization.
  2. 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) [∴ Am x An = (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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NCERT Class 9 Maths Lab Manual