**CBSE Class 9 Maths Lab Manual – Algebraic Identity (a + b)**^{3} = a^{3} + b^{3} + 3a^{2}b + 3ab^{2}

^{3}= a

^{3}+ b

^{3}+ 3a

^{2}b + 3ab

^{2}

**Objective**

To verify the identity (a+b)^{3} = a^{3} + b^{3} + 3a^{2}b + 3ab^{2} geometrically by using sets of unit cubes.

**Prerequisite Knowledge**

- Volume of a cube = (edge)
^{3} - 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} = a^{3} + b^{3} + 3a^{2}b + 3ab^{2}, we shall take value of a = 3units and value of b = 1unit.

- 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).

- 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 b
^{3}[fig (v)].

- 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).

**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:

- Volume of cube in fig. (ii) = a
^{3} - Volume of cuboids in fig.(iii) = 3(a)(a)(b) = 3ba
^{2} - Volume of cuboids in fig. (iv) = 3(a)(b)(b) = 3ab
^{2} - Volume of cube in fig. (v) = b
^{3}

Total volume of cube in fig. (i) = Total volume of cubes and cuboids in fig. (ii), (iii), (iv) and (v)

(a+b)^{3} = a^{3} + b^{3} + 3a^{2}b + 3ab^{2}

Hence, this identity is verified geometrically.

**Result**

The identity (a+b)^{3} = a^{3} + b^{3} + 3a^{2}b + 3ab^{2} 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 11^{3} by using the formula (a+b)^{3} = a^{3} + b^{3} + 3a^{2}b + 3ab^{2}.

**Answer:**

(10 + 1)^{3 }= (10)^{3} + 3 x (10)^{2} x 1 + 3 x 10 x 1 + (1)^{3} = 1331.

**Question 2.**

Find 23^{3} by using the formula (a+b)^{3} = a^{3} + b^{3} + 3a^{2}b + 3ab^{2}

**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:**

27x^{3} + 125y^{3} + 135x^{2}y + 225xy^{2}.

**Question 4.**

Find (10 + 2)^{3}.

**Answer:**

10^{3} + 2^{3} +600+ 120 = 1728.

**Question 5.**

In the activity of (a + b)^{3}, what do you mean by 3a^{2}b, 3ab^{2}?

**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 m^{3} 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) x^{3} + \(\frac { 8 }{ 27 }\) y^{3} + 2x^{2}y + \(\frac { 4 }{ 3 }\) xy^{2}

(ii) x^{3} + \(\frac { 8 }{ 27 }\) y^{3}

(iii) x^{3} + \(\frac { 2 }{ 3 }\) y^{3}

(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 + 125a^{3} + 135a+ 225a^{3}:

(i) (3 – 5a)^{3}

(ii) (3 + 5a)^{3}

(iii) 27 + 125a^{3}

(iv) none of these

**Question 4.**

Simplify: (\(\frac { 3 }{ 2 }\) x + 5)^{3}

(i) \(\frac { 27 }{ 8 }\) x^{3} + 125 + \(\frac { 135 }{ 4 }\) x^{2} + \(\frac { 225 }{ 2 }\) x

(ii) \(\frac { 27 }{ 8 }\) x^{3} + 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 23a^{3} + 8 = 8x^{3} – 16x^{2} + 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) 1^{3}

(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 + x^{3} + 6x(2 + x)

(ii) 4 + x^{2} + 8x

(iii) x^{2} + 4 + 4x

(iv) none of these

**Answers**

- (i)
- (i)
- (ii)
- (i)
- (ii)
- (ii)
- (i)
- (iii)
- (iii)
- (i)

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