CBSE Class 9 Maths Lab Manual – Algebraic Identity (a + b)3 = a3 + b3 + 3a2b + 3ab2

CBSE Class 9 Maths Lab Manual – Algebraic Identity (a + b)³ = a³ + b³ + 3a²b + 3ab²

Objective
To verify the identity (a+b)³ = a³ + b³ + 3a²b + 3ab² geometrically by using sets of unit cubes.

Prerequisite Knowledge

1. Volume of a cube = (edge)³
2. Volume of a cuboid = l x b x h

Materials Required
A set of 128 plastic cubes or wooden cubes with dimensions (1unit x 1unit x 1unit).

Procedure
To verify the identity (a+b)³ = a³ + b³ + 3a²b + 3ab², we shall take value of a = 3units and value of b = 1unit.

1. First we will make a cube of dimensions (a+b) i.e., (3+1 = 4 units). For this, we will use 64 unit cubes and arrange them as shown in fig.(i).
   
2. From other set of 64 cubes we will make arrangements as shown in figures.
   • Arrange 27 cubes such that a x a x a cube is formed i.e., 3 x 3 x 3 set is formed in fig.(ii).
     
   • Arrange 9 cubes such that 3 columns of 3 cubes is formed as shown in fig. (iii). Make 3 sets of such arrangement.
     
   • Arrange 3 cubes such that 1 column of 3 cubes is formed as shown in fig. (iv). Make 3 sets of such arrangement.
     
   • Last arrangement consists of only 1 cube of volume b³ [fig (v)].
     

Observation and Calculation
In fig. (i) we have used 64 cubes i.e., (a+b)³
Other set of 64 cubes are arranged in following manner:

1. Volume of cube in fig. (ii) = a³
2. Volume of cuboids in fig.(iii) = 3(a)(a)(b) = 3ba²
3. Volume of cuboids in fig. (iv) = 3(a)(b)(b) = 3ab²
4. Volume of cube in fig. (v) = b³

Total volume of cube in fig. (i) = Total volume of cubes and cuboids in fig. (ii), (iii), (iv) and (v)
(a+b)³ = a³ + b³ + 3a²b + 3ab²
Hence, this identity is verified geometrically.

Result
The identity (a+b)³ = a³ + b³ + 3a²b + 3ab² is verified geometrically by using cubes and cuboids.

Learning Outcome
In this way, students can learn the concept of verifying the identity geometrically by adding volume of cubes and cuboids.

Activity Time
Students must perform this activity for any other values of a and b, e.g., a=4, b=1 and find volumes of different cubes and cuboids used in this activity.

Viva Voce

Question 1.
Find 11³ by using the formula (a+b)³ = a³ + b³ + 3a²b + 3ab².
Answer:
(10 + 1)³ = (10)³ + 3 x (10)² x 1 + 3 x 10 x 1 + (1)³ = 1331.

Question 2.
Find 23³ by using the formula (a+b)³ = a³ + b³ + 3a²b + 3ab²
Answer:
(20 + 3)³ = (20)³ + (3)³ + 3.(20)².(3) + 3.(20)(3)²
= 8000 + 27 + 9(400) + 60(9) = 8000 + 27 + 3600 + 540 = 12167.

Question 3.
Find (5x + 5y)³
Answer:
27x³ + 125y³ + 135x²y + 225xy².

Question 4.
Find (10 + 2)³.
Answer:
10³ + 2³ +600+ 120 = 1728.

Question 5.
In the activity of (a + b)³, what do you mean by 3a²b, 3ab²?
Answer:
3 set of cuboids with volumes a x a x b and 3 sets of cuboids with volumes a x b x b.

Question 6.
If one side of a cube is (a – b), then what is the volume of the cube ?
Answer:
Volume of cube = (a – b)³.

Question 7.
Write the degree of (x + 2y)³
Answer:
3.

Question 8.
Write the coefficient of m³ in (2m + n)³.
Answer:
12n

Question 9.
Is (a + b)³ binomial ?
Answer:
No

Question 10.
Write the integeral zeroes of (x+8)³
Answer:
-8

Multiple Choice Questions

Question 1.
Write in expanded form: (x + \(\frac { 2 }{ 3 }\) y)³
(i) x³ + \(\frac { 8 }{ 27 }\) y³ + 2x²y + \(\frac { 4 }{ 3 }\) xy²
(ii) x³ + \(\frac { 8 }{ 27 }\) y³
(iii) x³ + \(\frac { 2 }{ 3 }\) y³
(iv) None of these

Question 2.
Evaluate using suitable identity (102)³:
(i) 1061208
(ii) 1062208
(iii) 10062208
(iv) none of these

Question 3.
Factorize: 27 + 125a³ + 135a+ 225a³:
(i) (3 – 5a)³
(ii) (3 + 5a)³
(iii) 27 + 125a³
(iv) none of these

Question 4.
Simplify: (\(\frac { 3 }{ 2 }\) x + 5)³
(i) \(\frac { 27 }{ 8 }\) x³ + 125 + \(\frac { 135 }{ 4 }\) x² + \(\frac { 225 }{ 2 }\) x
(ii) \(\frac { 27 }{ 8 }\) x³ + 125
(iii) \(\frac { 27 }{ 8 }\) x + 125
(iv) none of these

Question 5.
Evaluate, (51)³ using identity (a + b)³:
(i) 133261
(ii) 132651
(iii) 133651
(iv) none of these

Question 6.
If x – 1 = a, is it true that 23a³ + 8 = 8x³ – 16x² + 16x – 8ax ?
(i) false
(ii) true
(iii) can’t say
(iv) none of these

Question 7.
Factorize: 1 x 1 x 1 + .02 x .02 x .02 + 3(.02)(1.02):
(i) (1.02)³
(ii) 1³
(iii) 0.023
(iv) none of these

Question 8.
Without multiplication, evaluate (1.01)³:
(i) 1.0303
(ii) 1.03301
(iii) 1.030301
(iv) none of these

Question 9.
If a3 + b3 = 152, a +b = 8, find value of (ab):
(i) 160
(ii) 14
(iii) 15
(iv) none of these

Question 10.
Write the product of (2 + x)(x + 2)² By using identity (a+b)³.
(i) 8 + x³ + 6x(2 + x)
(ii) 4 + x² + 8x
(iii) x² + 4 + 4x
(iv) none of these

Answers

1. (i)
2. (i)
3. (ii)
4. (i)
5. (ii)
6. (ii)
7. (i)
8. (iii)
9. (iii)
10. (i)

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