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
- 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) [∴ 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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