Selina Solutions Concise Mathematics Class 6 Chapter 19 Fundamental Operations has accurate answers, designed by subject matter experts at BYJUâ€™S, in accordance with the grasping abilities of students. The solutions play a key role in improving studentsâ€™ problem solving abilities in a short duration. Those who aim to achieve high marks in the annual examination are suggested to solve the textbook questions using these solutions. In order to obtain more conceptual knowledge, students can download Selina Solutions Concise Mathematics Class 6 Chapter 19 Fundamental Operations PDF, from the below mentioned links.

Chapter 19 provides in-depth knowledge relying on Fundamental Operations such as addition, subtraction, multiplication and division. Students who practice these solutions on a regular basis, learn multiple ways of solving complex problems effortlessly.

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### Exercises of Selina Solutions Concise Mathematics Class 6 Chapter 19: Fundamental Operations

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Exercise 19(A)

**1. Fill in the blanks:**

**(i) 5 + 4 = â€¦â€¦.. and 5x + 4x = â€¦â€¦â€¦**

**(ii) 12 + 18 = â€¦â€¦. and 12x ^{2}y + 18x^{2}y = â€¦â€¦..**

**(iii) 7 + 16 = â€¦â€¦.. and 7a + 16b = â€¦â€¦.**

**(iv) 1 + 3 = â€¦â€¦. and x ^{2}y + 3xy^{2} = â€¦â€¦**

**(v) 7 â€“ 4 = â€¦â€¦ and 7ab â€“ 4ab = â€¦â€¦â€¦**

**Solution:**

(i) 5 + 4 = **9 **and 5x + 4x = **9x**

(ii) 12 + 18 = **30** and 12x^{2}y + 18x^{2}y = **30x ^{2}y**

(iii) 7 + 16 = **23** and 7a + 16b = **7a + 16b**

(iv) 1 + 3 = **4 **and x^{2}y + 3xy^{2} = **x ^{2}y + 3xy^{2}**

(v) 7 â€“ 4 = **3 **and 7ab â€“ 4ab = **3ab**

**2. Fill in the blanks:**

**(i) The sum of -2 and -5 = â€¦â€¦â€¦ and the sum of -2x and -5x = â€¦â€¦â€¦.**

**(ii) The sum of 8 and -3 = â€¦â€¦â€¦ and the sum of 8ab and -3ab = â€¦â€¦â€¦â€¦.**

**(iii) The sum of -15 and -4 = â€¦â€¦â€¦ and the sum of -15x and -4y = â€¦â€¦â€¦..**

**(iv) 15 + 8 + 3 = â€¦â€¦.. and 15x + 8y + 3x = â€¦â€¦â€¦â€¦**

**(v) 12 â€“ 9 + 15 = â€¦â€¦â€¦â€¦ and 12ab â€“ 9ab + 15ba = â€¦â€¦â€¦.**

**Solution:**

(i) The sum of -2 and -5 = **– 7** and the sum of -2x and -5x = **– 7x**

(ii) The sum of 8 and -3 = **5 **and the sum of 8ab and -3ab = **5ab**

(iii) The sum of -15 and -4 = **– 19** and the sum of -15x and -4y = **– 15x â€“ 4y**

(iv) 15 + 8 + 3 = **26 **and 15x + 8y + 3x = **18x + 8y**

(v) 12 â€“ 9 + 15 = **18** and 12ab â€“ 9ab + 15ba = **18ab**

**3. Add:**

**(i) 8xy and 3xy**

**(ii) 2xyz, xyz and 6xyz**

**(iii) 2a, 3a and 4b**

**(iv) 3x and 2y**

**(v) 5m, 3n and 4p**

**Solution:**

(i) 8xy and 3xy

The addition of 8xy and 3xy is calculated as follows

8xy + 3xy = 11xy

(ii) 2xyz, xyz and 6xyz

The addition of 2xyz, xyz and 6xyz is calculated as follows

2xyz + xyz + 6xyz = 9xyz

(iii) 2a, 3a and 4b

The addition of 2a, 3a and 4b is calculated as follows

2a + 3a + 4b = 5a + 4b

(iv) 3x and 2y

The addition of 3x and 2y is calculated as follows

3x + 2y = 3x + 2y

(v) 5m, 3n and 4p

The addition of 5m, 3n and 4p is calculated as follows

5m + 3n + 4p = 5m + 3n + 4p

**4. Evaluate:**

**(i) 6a â€“ a â€“ 5a â€“ 2a**

**(ii) 2b â€“ 3b â€“ b + 4b**

**(iii) 3x â€“ 2x â€“ 4x + 7x**

**(iv) 5ab + 2ab â€“ 6ab + ab**

**(v) 8x â€“ 5y â€“ 3x + 10y**

**Solution:**

(i) 6a â€“ a â€“ 5a â€“ 2a

The value of given expression is calculated as below

6a â€“ a â€“ 5a â€“ 2a = (6 â€“ 1 â€“ 5 â€“ 2) a

We get,

= (5 â€“ 5 â€“ 2) a

= – 2a

Therefore, 6a â€“ a â€“ 5a â€“ 2a = – 2a

(ii) 2b â€“ 3b â€“ b + 4b

The value of given expression is calculated as below

2b â€“ 3b â€“ b + 4b = 2b + 4b â€“ (3 + 1) b

We get,

= 6b â€“ 4b

= 2b

Therefore, 2b â€“ 3b â€“ b + 4b = 2b

(iii) 3x â€“ 2x â€“ 4x + 7x

The given expression is calculated as below

3x â€“ 2x â€“ 4x + 7x = 3x + 7x â€“ 2x â€“ 4x

= (3 + 7) x â€“ (2 + 4) x

= 10x â€“ 6x

= 4x

Therefore, 3x â€“ 2x â€“ 4x + 7x = 4x

(iv) 5ab + 2ab â€“ 6ab + ab

The given expression is calculated as below

5ab + 2ab â€“ 6ab + ab = 5ab + 2ab + ab â€“ 6ab

We get,

= 8ab â€“ 6ab

= 2ab

Therefore, 5ab + 2ab â€“ 6ab + ab = 2ab

(v) 8x â€“ 5y â€“ 3x + 10y

The given expression is calculated as below

8x â€“ 5y â€“ 3x + 10y = 8x â€“ 3x + 10y â€“ 5y

= 5x + 5y

Therefore, 8x â€“ 5y â€“ 3x + 10y = 5x + 5y

**5. Evaluate:**

**(i) -7x + 9x + 2x â€“ 2x**

**(ii) 5ab â€“ 2ab â€“ 8ab + 6ab**

**(iii) -8a â€“ 3a + 12a + 13a â€“ 6a**

**(iv) 19abc â€“ 11abc â€“ 12abc + 14abc**

**Solution:**

(i) -7x + 9x + 2x â€“ 2x

The value of given expression is calculated as follows

-7x + 9x + 2x â€“ 2x = 9x + 2x â€“ 7x â€“ 2x

= 11x â€“ 9x

We get,

= 2x

Hence, -7x + 9x + 2x â€“ 2x = 2x

(ii) 5ab â€“ 2ab â€“ 8ab + 6ab

The value of given expression is calculated as follows

5ab â€“ 2ab â€“ 8ab + 6ab = 5ab + 6ab â€“ 2ab â€“ 8ab

We get,

= 11ab â€“ 10ab

= ab

Hence, 5ab â€“ 2ab â€“ 8ab + 6ab = ab

(iii) **–**8a â€“ 3a + 12a + 13a â€“ 6a

The value of given expression is calculated as follows

-8a â€“ 3a + 12a + 13a â€“ 6a = 12a + 13a â€“ (8a + 3a + 6a)

= 25a â€“ 17a

= 8a

Hence, -8a â€“ 3a + 12a + 13a â€“ 6a = 8a

(iv) 19abc â€“ 11abc â€“ 12abc + 14abc

The value of given expression is calculated as follows

19abc â€“ 11abc â€“ 12abc + 14abc = abc (19 â€“ 11 â€“ 12 + 14)

= abc (33 â€“ 23)

= 10abc

Hence, 19abc â€“ 11abc â€“ 12abc + 14abc = 10abc

**6. Subtract the first term from the second:**

**(i) 4ab, 6ba**

**(ii) 4.8b, 6.8b**

**(iii) 3.5abc, 10.5abc**

**(iv) 3(1 / 2) mn, 8(1 / 2)nm**

**Solution:**

(i) 4ab, 6ba

The subtraction of first term from the second term is calculated as below

6ba â€“ 4ab = 2ab

(ii) 4.8b, 6.8b

The subtraction of first term from the second term is calculated as below

6.8b â€“ 4.8b = 2b

(iii) 3.5abc, 10.5abc

The subtraction of first term from the second term is calculated as below

10.5abc â€“ 3.5abc = 7abc

(iv) 3(1 / 2) mn, 8(1 / 2)nm

The subtraction of first term from the second term is calculated as below

8(1 / 2)nm â€“ 3 (1 / 2) mn = (17 / 2)nm â€“ (7 / 2)mn

We get,

= [(17mn â€“ 7mn) / 2]

= (10 / 2)mn

= 5mn

**7. Simplify:**

**(i) 2a ^{2}b^{2} + 5ab^{2} + 8a^{2}b^{2} â€“ 3ab^{2}**

**(ii) 4a + 3b â€“ 2a â€“ b**

**(iii) 2xy + 4yz + 5xy + 3yz â€“ 6xy**

**(iv) ab + 15ab â€“ 11ab â€“ 2ab**

**(v) 6a ^{2} â€“ 3b^{2} + 2a^{2} + 5b^{2} â€“ 4a^{2}**

**Solution:**

(i) 2a^{2}b^{2} + 5ab^{2} + 8a^{2}b^{2} â€“ 3ab^{2}

The simplified form of the given expression is calculated as follows

2a^{2}b^{2} + 5ab^{2} + 8a^{2}b^{2} â€“ 3ab^{2} = 2a^{2}b^{2} + 8a^{2}b^{2} + 5ab^{2} â€“ 3ab^{2}

We get,

= 10a^{2}b^{2} + 2ab^{2}

Therefore, 2a^{2}b^{2} + 5ab^{2} + 8a^{2}b^{2} â€“ 3ab^{2 }= 10a^{2}b^{2} + 2ab^{2}

(ii) 4a + 3b â€“ 2a â€“ b

The simplified form of the given expression is calculated as follows

4a + 3b â€“ 2a â€“ b = 4a â€“ 2a + 3b â€“ b

= 2a + 2b

Therefore, 4a + 3b â€“ 2a â€“ b = 2a + 2b

(iii) 2xy + 4yz + 5xy + 3yz â€“ 6xy

The simplified form of the given expression is calculated as follows

2xy + 4yz + 5xy + 3yz â€“ 6xy = 2xy + 5xy â€“ 6xy + 4yz + 3yz

= xy + 7yz

Therefore, 2xy + 4yz + 5xy + 3yz â€“ 6xy = xy + 7yz

(iv) ab + 15ab â€“ 11ab â€“ 2ab

The simplified form of the given expression is calculated as follows

ab + 15ab â€“ 11ab â€“ 2ab = 16ab â€“ 13ab

= 3ab

Therefore, ab + 15ab â€“ 11ab â€“ 2ab = 3ab

(v) 6a^{2} â€“ 3b^{2} + 2a^{2} + 5b^{2} â€“ 4a^{2}

The simplified form of the given expression is calculated as follows

6a^{2} â€“ 3b^{2} + 2a^{2} + 5b^{2} â€“ 4a^{2} = 6a^{2} + 2a^{2} â€“ 4a^{2} + 5b^{2} â€“ 3b^{2}

We get,

= 4a^{2} + 2b^{2}

Therefore, 6a^{2} â€“ 3b^{2} + 2a^{2} + 5b^{2} â€“ 4a^{2} = 4a^{2} + 2b^{2}

Exercise 19(B)

**1. Find the sum of:**

**(i) 3a + 4b + 7c, – 5a + 3b â€“ 6c**

**and 4a â€“ 2b â€“ 4c**

**(ii) 2x ^{2} + xy â€“ y^{2}, – x^{2} + 2xy + 3y^{2}**

**and 3x ^{2} â€“ 10xy + 4y^{2}**

**(iii) x ^{2} â€“ x + 1, -5x^{2} + 2x â€“ 2**

**and 3x ^{2} â€“ 3x + 1**

**(iv) a ^{2} â€“ ab + bc, 2ab + bc â€“ 2a^{2}**

**and – 3bc + 3a ^{2} + ab**

**(v) 4x ^{2} + 7 â€“ 3x, 4x â€“ x^{2} + 8**

**and â€“ 10 + 5x â€“ 2x ^{2}**

**Solution:**

(i) 3a + 4b + 7c, – 5a + 3b â€“ 6c

and 4a â€“ 2b â€“ 4c

The sum of 3a + 4b + 7c, – 5a + 3b â€“ 6c and 4a â€“ 2b â€“ 4c is calculated as shown below

(3a + 4b + 7c) + (-5a + 3b â€“ 6c) + (4a â€“ 2b â€“ 4c)

= 3a + 4a â€“ 5a + 4b + 3b â€“ 2b + 7c â€“ 6c â€“ 4c

We get,

= 2a +5b â€“ 3c

Hence, the sum of 3a + 4b + 7c, – 5a + 3b â€“ 6c and 4a â€“ 2b â€“ 4c is 3c

(ii) 2x^{2} + xy â€“ y^{2}, – x^{2} + 2xy + 3y^{2}

and 3x^{2} â€“ 10xy + 4y^{2}

The sum of 2x^{2} + xy â€“ y^{2}, – x^{2} + 2xy + 3y^{2} and 3x^{2} â€“ 10xy + 4y^{2} is calculated as shown below

(2x^{2} + xy â€“ y^{2}) + (- x^{2} + 2xy + 3y^{2}) + (3x^{2} â€“ 10xy + 4y^{2})

= 2x^{2} â€“ x^{2} + 3x^{2} + xy + 2xy â€“ 10xy + 3y^{2} + 4y^{2} â€“ y^{2}

We get,

= 4x^{2} â€“ 7xy + 6y^{2}

Hence, the sum of 2x^{2} + xy â€“ y^{2}, – x^{2} + 2xy + 3y^{2} and 3x^{2} â€“ 10xy + 4y^{2} is 4x^{2} â€“ 7xy + 6y^{2}

(iii) x^{2} â€“ x + 1, -5x^{2} + 2x â€“ 2 and 3x^{2} â€“ 3x + 1

The sum of (x^{2} â€“ x + 1), (- 5x^{2} + 2x â€“ 2) and (3x^{2} â€“ 3x + 1) is calculated as shown below

(x^{2} â€“ x + 1) + (- 5x^{2} + 2x â€“ 2) + (3x^{2} â€“ 3x + 1)

= x^{2} â€“ 5x^{2} + 3x^{2} + 2x â€“ x â€“ 3x + 1 +1 â€“ 2

= – x^{2} â€“ 2x

Hence, the sum of (x^{2} â€“ x + 1), (- 5x^{2} + 2x â€“ 2) and (3x^{2} â€“ 3x + 1) is – x^{2} â€“ 2x

(iv) a^{2} â€“ ab + bc, 2ab + bc â€“ 2a^{2} and – 3bc + 3a^{2} + ab

The sum of (a^{2} â€“ ab + bc), (2ab + bc â€“ 2a^{2}) and (- 3bc + 3a^{2} + ab) is calculated as shown below

(a^{2} â€“ ab + bc) + (2ab + bc â€“ 2a^{2}) + (- 3bc + 3a^{2} + ab)

= a^{2} â€“ 2a^{2} + 3a^{2} + 2ab + ab â€“ ab + bc + bc â€“ 3bc

We get,

= 2a^{2} +2ab – bc

Hence, the sum of (a^{2} â€“ ab + bc), (2ab + bc â€“ 2a^{2}) and (- 3bc + 3a^{2} + ab) is 2a^{2} +2ab – bc

(v) 4x^{2} + 7 â€“ 3x, 4x â€“ x^{2} + 8 and â€“ 10 + 5x â€“ 2x^{2}

The sum of (4x^{2} + 7 â€“ 3x), (4x â€“ x^{2} + 8) and (- 10 + 5x â€“ 2x^{2}) is calculated as shown below

(4x^{2} + 7 â€“ 3x) + (4x â€“ x^{2} + 8) + (- 10 + 5x â€“ 2x^{2})

= 4x^{2} â€“ x^{2} â€“ 2x^{2} + 7 + 8 â€“ 10 + 4x + 5x â€“ 3x

We get,

= x^{2} + 5 + 6x

Hence, the sum of (4x^{2} + 7 â€“ 3x), (4x â€“ x^{2} + 8) and (- 10 + 5x â€“ 2x^{2}) is x^{2} + 5 + 6x

**2.Add the following expressions:**

**(i) â€“ 17x ^{2} â€“ 2xy + 23y^{2}, – 9y^{2} + 15x^{2} + 7xy**

**and 13x ^{2} + 3y^{2} â€“ 4xy**

**(ii) â€“ x ^{2} â€“ 3xy + 3y^{2} + 8, 3x^{2} â€“ 5y^{2} â€“ 3 + 4xy**

**and â€“ 6xy + 2x ^{2} â€“ 2 + y^{2}**

**(iii) a ^{3} â€“ 2b^{3} + a, b^{3} â€“ 2a^{3} + b**

**and â€“ 2b + 2b ^{3} â€“ 5a + 4a^{3}**

**Solution:**

(i) The sum of (â€“ 17x^{2} â€“ 2xy + 23y^{2}), (- 9y^{2} + 15x^{2} + 7xy) and (13x^{2} + 3y^{2} â€“ 4xy) is calculated as follows

(â€“ 17x^{2} â€“ 2xy + 23y^{2}) + (- 9y^{2} + 15x^{2} + 7xy) + (13x^{2} + 3y^{2} â€“ 4xy)

= – 17x^{2} + 15x^{2} + 13x^{2} â€“ 2xy â€“ 4xy + 7xy + 23y^{2} + 3y^{2} â€“ 9y^{2}

We get,

= 11x^{2} + xy + 17y^{2}

Therefore, the sum of (â€“ 17x^{2} â€“ 2xy + 23y^{2}), (- 9y^{2} + 15x^{2} + 7xy) and (13x^{2} + 3y^{2} â€“ 4xy) is 11x^{2} + xy + 17y^{2}

(ii) â€“ x^{2} â€“ 3xy + 3y^{2} + 8, 3x^{2} â€“ 5y^{2} â€“ 3 + 4xy and â€“ 6xy + 2x^{2} â€“ 2 + y^{2}

The sum of (â€“ x^{2} â€“ 3xy + 3y^{2} + 8), (3x^{2} â€“ 5y^{2} â€“ 3 + 4xy) and (â€“ 6xy + 2x^{2} â€“ 2 + y^{2}) is calculated as follows

(â€“ x^{2} â€“ 3xy + 3y^{2} + 8) + (3x^{2} â€“ 5y^{2} â€“ 3 + 4xy) + (â€“ 6xy + 2x^{2} â€“ 2 + y^{2})

= – x^{2} + 3x^{2} + 2x^{2} â€“ 3xy â€“ 6xy + 4xy + 3y^{2} + y^{2} â€“ 5y^{2} + 8 â€“ 3 â€“ 2

We get,

= 4x^{2} â€“ 5xy â€“ y^{2} + 3

Therefore, the sum of (â€“ x^{2} â€“ 3xy + 3y^{2} + 8), (3x^{2} â€“ 5y^{2} â€“ 3 + 4xy) and (â€“ 6xy + 2x^{2} â€“ 2 + y^{2}) is 4x^{2} â€“ 5xy â€“ y^{2} + 3

(iii) a^{3} â€“ 2b^{3} + a, b^{3} â€“ 2a^{3} + b and â€“ 2b + 2b^{3} â€“ 5a + 4a^{3}

The sum of (a^{3} â€“ 2b^{3} + a), (b^{3} â€“ 2a^{3} + b) and (â€“ 2b + 2b^{3} â€“ 5a + 4a^{3}) is calculated as follows

(a^{3} â€“ 2b^{3} + a) + (b^{3} â€“ 2a^{3} + b) + (â€“ 2b + 2b^{3} â€“ 5a + 4a^{3})

= a^{3} + 4a^{3} â€“ 2a^{3} â€“ 2b^{3} + b^{3} + 2b^{3} + a â€“ 5a + b â€“ 2b

We get,

= 3a^{3} + b^{3} â€“ 4a â€“ b

Therefore, the sum of (a^{3} â€“ 2b^{3} + a), (b^{3} â€“ 2a^{3} + b) and (â€“ 2b + 2b^{3} â€“ 5a + 4a^{3}) is 3a^{3} + b^{3} â€“ 4a â€“ b

**3. Evaluate:**

**(i) 3a â€“ (a + 2b)**

**(ii) (5x â€“ 3y) â€“ (x + y)**

**(iii) (8a + 15b) â€“ (3b â€“ 7a)**

**(iv) (8x + 7y) â€“ (4y â€“ 3x)**

**(v) 7 â€“ (4a â€“ 5)**

**Solution:**

(i) 3a â€“ (a + 2b)

The value of the given expression is calculated as below

3a â€“ (a + 2b)

= 3a â€“ a â€“ 2b

= 2a â€“ 2b

Taking 2 as common, we get

= 2(a â€“ b)

(ii) (5x â€“ 3y) â€“ (x + y)

The value of the given expression is calculated as below

(5x â€“ 3y) â€“ (x + y)

= 5x â€“ x â€“ 3y â€“ y

= 4x â€“ 4y

Taking 4 as common, we get

= 4(x â€“ y)

(iii) (8a + 15b) â€“ (3b â€“ 7a)

The value of the given expression is calculated as below

(8a + 15b) â€“ (3b â€“ 7a)

= 8a + 7a + 15b â€“ 3b

On calculation, we get

= 15a + 12b

(iv) (8x + 7y) â€“ (4y â€“ 3x)

The value of the given expression is calculated as below

(8x + 7y) â€“ (4y â€“ 3x)

= 8x + 3x + 7y â€“ 4y

On further calculation, we get

= 11x + 3y

(v) 7 â€“ (4a â€“ 5)

The value of the given expression is calculated as below

7 â€“ (4a â€“ 5)

= 7 â€“ 4a + 5

We get,

= 12 â€“ 4a

**4. Subtract:**

**(i) 5a â€“ 3b + 2c from a â€“ 4b â€“ 2c**

**(ii) 4x â€“ 6y + 3z from 12x + 7y â€“ 21z**

**(iii) 5 â€“ a â€“ 4b + 4c from 5a â€“ 7b + 2c**

**(iv) â€“ 8x â€“ 12y + 17z from x â€“ y â€“ z**

**(v) 2ab + cd â€“ ac â€“ 2bd from ab â€“ 2cd + 2ac + bd**

**Solution:**

(i) 5a â€“ 3b + 2c from a â€“ 4b â€“ 2c

The value of the subtraction is calculated as follows

(a â€“ 4b â€“ 2c) â€“ (5a â€“ 3b + 2c)

= a â€“ 5a â€“ 4b + 3b â€“ 2c â€“ 2c

We get,

= – 4a â€“ b â€“ 4c

(ii) 4x â€“ 6y + 3z from 12x + 7y â€“ 21z

The value of the subtraction is calculated as follows

(12x + 7y â€“ 21z) â€“ (4x â€“ 6y + 3z)

= 12x â€“ 4x + 7y + 6y â€“ 21z â€“ 3z

On further calculation, we get

= 8x + 13y â€“ 24z

(iii) 5 â€“ a â€“ 4b + 4c from 5a â€“ 7b + 2c

The value of the subtraction is calculated as follows

(5a â€“ 7b + 2c) â€“ (5 â€“ a â€“ 4b + 4c)

= 5a + a â€“ 7b + 4b + 2c â€“ 4c â€“ 5

We get,

=6a â€“ 3b â€“ 2c â€“ 5

(iv) â€“ 8x â€“ 12y + 17z from x â€“ y â€“ z

The value of the subtraction is calculated as follows

(x â€“ y â€“ z) â€“ (- 8x â€“ 12y + 17z)

= x + 8x + 12y â€“ y â€“ z â€“ 17z

We get,

= 9x + 11y â€“ 18z

(v) 2ab + cd â€“ ac â€“ 2bd from ab â€“ 2cd + 2ac + bd

The value of the subtraction is calculated as follows

(ab â€“ 2cd + 2ac + bd) â€“ (2ab + cd â€“ ac â€“ 2bd)

= ab â€“ 2ab â€“ 2cd â€“ cd + 2ac + ac + bd + 2bd

On calculating further, we get

= – ab â€“ 3cd + 3ac + 3bd

**5. **

**(i) Take â€“ ab + bc â€“ ca from bc â€“ ca + ab.**

**(ii) Take 5x + 6y â€“ 3z from 3x + 5y â€“ 4z.**

**(iii) Take (-3 / 2) p + q â€“ r from (1 / 2)p â€“ (1 / 3)q â€“ (3 / 2) r**

**(iv) Take 1 â€“ a + a ^{2} from a^{2} + a + 1**

**Solution:**

(i) The value of the subtraction is calculated as,

(bc â€“ ca + ab) â€“ (-ab + bc â€“ ca)

= bc â€“ bc â€“ ca + ca + ab + ab

We get,

= 2ab

Hence, (bc â€“ ca + ab) â€“ (-ab + bc â€“ ca) = 2ab

(ii) The value of the subtraction is calculated as,

(3x + 5y â€“ 4z) â€“ (5x + 6y â€“ 3z)

= 3x â€“ 5x + 5y â€“ 6y â€“ 4z + 3z

On simplification, we get

= – 2x â€“ y â€“ z

Hence, (3x + 5y â€“ 4z) â€“ (5x + 6y â€“ 3z) = – 2x â€“ y â€“ z

(iii) The value of the subtraction is calculated as,

[(1 / 2)p â€“ (1 / 3)q â€“ (3 / 2) r] â€“ [(-3 / 2) p + q â€“ r]= (1 / 2)p + (3 / 2)p â€“ (1 / 3)q â€“ q â€“ (3 / 2)r + r

On further calculation, we get

= 2p â€“ (4 / 3)q â€“ (1 / 2)r

Hence, [(1 / 2)p â€“ (1 / 3)q â€“ (3 / 2) r] â€“ [(-3 / 2) p + q â€“ r] = 2p â€“ (4 / 3)q â€“ (1 / 2)r

(iv) The value of the subtraction is calculated as,

(a^{2} + a + 1) â€“ (1 â€“ a + a^{2})

= a^{2} â€“ a^{2} + a + a + 1 â€“ 1

We get,

= a + a

= 2a

**6. From the sum of x + y â€“ 2z and 2x â€“ y + z subtract x + y + z.**

**Solution:**

The value of terms as per the question is calculated as follows

(x + y â€“ 2z) + (2x â€“ y + z) – (x + y + z)

= x + 2x â€“ x + y â€“ y â€“ y â€“ 2z â€“ z + z

We get,

= 2x â€“ y â€“ 2z

Therefore, (x + y â€“ 2z) + (2x â€“ y + z) – (x + y + z) = 2x â€“ y â€“ 2z

**7. From the sum of 3a â€“ 2b + 4c and 3b â€“ 2c subtract a â€“ b â€“ c.**

**Solution:**

The value of terms as per the question is calculated as shown below

(3a â€“ 2b + 4c) + (3b â€“ 2c) â€“ (a â€“ b â€“ c)

= 3a â€“ 2b + 4c + 3b â€“ 2c â€“ a + b + c

On further calculation, we get

= 3a â€“ a + 3b + b â€“ 2b + 4c + c â€“ 2c

= 2a + 2b + 3c

Hence, (3a â€“ 2b + 4c) + (3b â€“ 2c) â€“ (a â€“ b â€“ c) = 2a + 2b + 3c

**8. Subtract x â€“ 2y â€“ z from the sum of 3x â€“ y + z and x + y â€“ 3z.**

**Solution:**

The value of terms as per the question is calculated as follows

(3x â€“ y + z) + (x + y â€“ 3z) â€“ (x â€“ 2y â€“ z)

= 3x + x â€“ x â€“ y + y + 2y + z â€“ 3z + z

We get,

= 3x + 2y â€“ z

Therefore, (3x â€“ y + z) + (x + y â€“ 3z) â€“ (x â€“ 2y â€“ z) = 3x + 2y â€“ z

**9. Subtract the sum of x + y and x â€“ z from the sum of x â€“ 2z and x + y + z**

**Solution: **

The value of terms as per the question is calculated as follows

(x â€“ 2z) + (x + y + z) â€“ {(x + y) + (x â€“ z)}

On further calculation, we get

= x + x â€“ x â€“x + y â€“ y + z + z â€“ 2z

We get,

= 0

Therefore, (x â€“ 2z) + (x + y + z) â€“ {(x + y) + (x â€“ z)} = 0

**10. By how much should x + 2y â€“ 3z be increased to get 3x?**

**Solution:**

The terms calculated as per the question is as follows

3x â€“ (x + 2y â€“ 3z)

= 3x â€“ x â€“ 2y + 3z

We get,

= 2x â€“ 2y + 3z

**11. The sum of two expressions is 5x ^{2} â€“ 3y^{2}. If one of them is 3x^{2} + 4xy â€“ y^{2}, find the other.**

**Solution:**

The other expression is calculated as follows

(5x^{2} â€“ 3y^{2}) â€“ (3x^{2} + 4xy â€“ y^{2})

= 5x^{2} â€“ 3x^{2} â€“ 4xy â€“ 3y^{2} + y^{2}

We get,

= 2x^{2} â€“ 4xy â€“ 2y^{2}

**12. The sum of two expressions is 3a ^{2} + 2ab â€“ b^{2}. If one of them is 2a^{2} + 3b^{2}, find the other.**

**Solution:**

The other expression is calculated as follows

(3a^{2} + 2ab â€“ b^{2}) â€“ (2a^{2} + 3b^{2})

On simplification, we get

=3a^{2} â€“ 2a^{2} â€“ b^{2} â€“ 3b^{2} + 2ab

= a^{2}– 4b^{2} + 2ab

Exercise 19(C)

**1. Fill in the blanks:**

**(i) 6 Ã— 3 = â€¦â€¦â€¦. and 6x Ã— 3x = â€¦â€¦â€¦â€¦**

**(ii) 6 Ã— 3 = â€¦â€¦… and 6x ^{2} Ã— 3x^{3} = â€¦â€¦â€¦â€¦**

**(iii) 5 Ã— 4 = â€¦â€¦â€¦. and 5x Ã— 4y = â€¦â€¦â€¦â€¦**

**(iv) 4 Ã— 7 = â€¦â€¦â€¦â€¦. and 4ax Ã— 7x = â€¦â€¦â€¦.**

**(v) 6 Ã— 2 = â€¦â€¦â€¦â€¦. and 6xy Ã— 2xy = â€¦â€¦….**

**Solution:**

(i) 6 Ã— 3 = 18

Hence,

6x Ã— 3x = 6 Ã— 3 Ã— x Ã— x

We get,

= 18 **Ã— **x^{2}

= 18x^{2}

Therefore, 6 Ã— 3 = 18 and 6x Ã— 3x = 18x^{2}

(ii) 6 Ã— 3 = 18

Hence,

6x^{2} Ã— 3x^{3} = 6 Ã— 3 Ã— x^{2 + 3}

= 18 **Ã— **x^{5}

= 18x^{5}

Therefore, 6 Ã— 3 = 18 and 6x^{2} Ã— 3x^{3 }= 18x^{5}

(iii) 5 Ã— 4 = 20 and 5x Ã— 4y = 5 Ã— 4 Ã— x Ã— y

= 20xy

Therefore, 5 Ã— 4 = 20 and 5x Ã— 4y = 20xy

(iv) 4 **Ã— 7 = **28

Hence,

4ax Ã— 7x = 4 **Ã— 7 Ã— **a Ã— x Ã— x

= 28 Ã— a Ã— x^{2}

= 28ax^{2}

Therefore, 4 Ã— 7 = 28 and 4ax Ã— 7x = 28ax^{2}

(v) 6 **Ã— **2 = 12

Hence,

6xy Ã— 2xy = 6 Ã— 2 Ã— x^{1 + 1} Ã— y^{1 + 1}

= 12 Ã— x^{2} Ã— y^{2}

= 12x^{2}y^{2}

Therefore, 6 Ã— 2 = 12 and 6xy Ã— 2xy = 12x^{2}y^{2}

**2. Fill in the blanks:**

**(i) 4x Ã— 6x Ã— 2 = â€¦â€¦â€¦**

**(ii) 3ab Ã— 6ax = â€¦â€¦â€¦â€¦**

**(iii) x Ã— 2x ^{2} Ã— 3x^{3} = â€¦â€¦â€¦**

**(iv) 5 Ã— 5a ^{3} = â€¦â€¦â€¦â€¦**

**(v) 6 Ã— 6x ^{2} Ã— 6x^{2}y^{2} = â€¦â€¦â€¦**

**Solution:**

(i) 4x Ã— 6x Ã— 2 = 4 Ã— 6 Ã— 2 Ã— x Ã— x

= 48 Ã— x^{2}

We get,

= 48x^{2}

Hence, 4x Ã— 6x Ã— 2 = 48x^{2}

(ii) 3ab **Ã— **6ax = 3 Ã— 6 Ã— a Ã— a Ã— b Ã— x

= 18 Ã— a^{2} Ã— b Ã— x

We get,

= 18a^{2}bx

Hence, 3ab Ã— 6ax = 18a^{2}bx

(iii) x **Ã— **2x^{2} Ã— 3x^{3} = 2 Ã— 3 Ã— x Ã— x^{2} Ã— x^{3}

= 6 Ã— x^{1 + 2 + 3}

= 6 Ã— x^{6}

= 6x^{6}

Hence, x Ã— 2x^{2} Ã— 3x^{3} = 6x^{6}

(iv) 5 Ã— 5a^{3} = 5 Ã— 5 Ã— a^{3}

= 25 Ã— a^{3}

We get,

= 25a^{3}

Hence, 5 Ã— 5a^{3} = 25a^{3}

(v) 6 **Ã— **6x^{2} Ã— 6x^{2}y^{2} = 6 Ã— 6 Ã— 6 Ã— x^{2} Ã— x^{2} Ã— y^{2}

= 216 Ã— x^{2+ 2} Ã— y^{2}

= 216 Ã— x^{4} Ã— y^{2}

We get,

= 216x^{4}y^{2}

Hence, 6 Ã— 6x^{2} Ã— 6x^{2}y^{2} = 216x^{4}y^{2}

**3. Find the value of:**

**(i) 3x ^{3} Ã— 5x^{4}**

**(ii) 5a ^{2} Ã— 7a^{7}**

**(iii) 3abc Ã— 6ac ^{3}**

**(iv) a ^{2}b^{2} Ã— 5a^{3}b^{4}**

**(v) 2x ^{2}y^{3} Ã— 5x^{3}y^{4}**

**Solution:**

(i) 3x^{3} Ã— 5x^{4}

3x^{3} Ã— 5x^{4} = 3 Ã— 5 Ã— x^{3} Ã— x^{4}

= 15 Ã— x^{3 + 4}

We get,

= 15 Ã— x^{7}

= 15x^{7}

Hence, the value of 3x^{3} Ã— 5x^{4} is 15x^{7}

(ii) 5a^{2} Ã— 7a^{7}

5a^{2} Ã— 7a^{7} = 5 Ã— 7 Ã— a^{2} Ã— a^{7}

= 35 Ã— a^{2 }^{+ 7}

= 35 Ã— a^{9}

We get,

= 35a^{9}

Hence, the value of 5a^{2} Ã— 7a^{7} is 35a^{9}

(iii) 3abc Ã— 6ac^{3}

3abc Ã— 6ac^{3} = 3 Ã— 6 Ã— a Ã— a Ã— b Ã— c Ã— c^{3}

=18 Ã— a^{1+ 1} Ã— b Ã— c^{1+3}

= 18 Ã— a^{2} Ã— b Ã— c^{4}

We get,

= 18a^{2}bc^{4}

Hence, the value of 3abc Ã— 6ac^{3} is 18a^{2}bc^{4}

(iv) a^{2}b^{2} Ã— 5a^{3}b^{4}

a^{2}b^{2} Ã— 5a^{3}b^{4} = 5 Ã— a^{2} Ã— a^{3} Ã— b^{2} Ã— b^{4}

= 5 Ã— a^{2+3} Ã— b^{2+4}

= 5 Ã— a^{5} Ã— b^{6}

We get,

= 5a^{5}b^{6}

Hence, the value of a^{2}b^{2} Ã— 5a^{3}b^{4} is 5a^{5}b^{6}

(v) 2x^{2}y^{3} Ã— 5x^{3}y^{4}

2x^{2}y^{3} Ã— 5x^{3}y^{4} = 2 Ã— 5 Ã— x^{2} Ã— x^{3} Ã— y^{3} Ã— y^{4}

= 10 Ã— x^{2+3} Ã— y^{3+4}

We get,

= 10 Ã— x^{5} Ã— y^{7}

= 10x^{5}y^{7}

Hence, the value of 2x^{2}y^{3} Ã— 5x^{3}y^{4} is 10x^{5}y^{7}

**4. Multiply:**

**(i) a + b by ab**

**(ii) 3ab â€“ 4b by 3ab**

**(iii) 2xy â€“ 5by by 4bx**

**(iv) 4x + 2y by 3xy**

**(v) 1 + 4x by x**

**Solution:**

(i) a + b by ab

The multiplication of a + b by ab is calculated as,

(a + b) Ã— ab = a Ã— ab + b Ã— ab

= a^{1+1}b + ab^{1+1}

We get,

= a^{2}b + ab^{2}

Hence, (a + b) by ab = a^{2}b + ab^{2}

(ii) 3ab â€“ 4b by 3ab

The multiplication of 3ab â€“ 4b by 3ab is calculated as,

(3ab â€“ 4b) Ã— 3ab = 3ab Ã— 3ab â€“ 4b Ã— 3ab

= 9a^{1+1}b^{1+1} â€“ 12ab^{1+1}

We get,

= 9a^{2}b^{2} â€“ 12ab^{2}

Therefore, (3ab â€“ 4b) by 3ab = 9a^{2}b^{2} â€“ 12ab^{2}

(iii) 2xy â€“ 5by by 4bx

The multiplication of 2xy â€“ 5by by 4bx is calculated as,

(2xy â€“ 5by) Ã— 4bx = 2xy Ã— 4bx â€“ 5by Ã— 4bx

= 8bx^{1+1}y â€“ 20b^{1+1}xy

We get,

= 8bx^{2}y â€“ 20b^{2}xy

Therefore, (2xy â€“ 5by) by 4bx = 8bx^{2}y â€“ 20b^{2}xy

(iv) 4x + 2y by 3xy

The multiplication of 4x + 2y by 3xy is calculated as,

(4x + 2y) Ã— 3xy = 4x Ã— 3xy + 2y Ã— 3xy

On simplification, we get

= 12x^{1+1}y + 6xy^{1+1}

= 12x^{2}y + 6xy^{2}

Therefore, (4x + 2y) by 3xy = 12x^{2}y + 6xy^{2}

(v) 1 + 4x by x

The multiplication of (1 + 4x) by x is calculated as,

(1 + 4x) Ã— x = 1 Ã— x + 4x Ã— x

On simplification, we get

= x + 4x^{1+1}

= x + 4x^{2}

Therefore, (1 + 4x) by x = x + 4x^{2}

**5. Multiply:**

**(i) â€“ x + y â€“ z and â€“ 2x**

**(ii) xy â€“ yz and x ^{2}yz^{2}**

**(iii) 2xyz + 3xy and â€“ 2y ^{2}z**

**(iv) â€“ 3xy ^{2} + 4x^{2}y and â€“ xy**

**(v) 4xy and â€“ x ^{2}y â€“ 3x^{2} y^{2}**

**Solution:**

(i) â€“ x + y â€“ z and â€“ 2x

The multiplication of the given expression is calculated as,

(- x + y â€“ z) Ã— – 2x = – x Ã— – 2x + y Ã— – 2x â€“ z Ã— – 2x

On further calculation, we get

= 2x^{1+1} â€“ 2xy + 2xz

= 2x^{2} â€“ 2xy + 2xz

Hence, the multiplication of (- x + y â€“ z) and â€“ 2x is 2x^{2} â€“ 2xy + 2xz

(ii) xy â€“ yz and x^{2}yz^{2}

The multiplication of the given expression is calculated as,

(xy â€“ yz) Ã— (x^{2}yz^{2}) = xy Ã— x^{2}yz^{2} â€“ yz Ã— x^{2}yz^{2}

We get,

= x^{1+2}y^{1+1}z^{2} â€“ x^{2}y^{1+1}z^{1+2}

= x^{3}y^{2}z^{2} â€“ x^{2}y^{2}z^{3}

Hence, the multiplication of (xy â€“ yz) and x^{2}yz^{2} = x^{3}y^{2}z^{2} â€“ x^{2}y^{2}z^{3}

(iii) 2xyz + 3xy and â€“ 2y^{2}z

The multiplication of the given expression is calculated as,

(2xyz + 3xy) Ã— – 2y^{2}z = 2xyz Ã— – 2y^{2}z + 3xy Ã— – 2y^{2}z

On further calculation, we get

= – 4xy^{1+2}z^{1+1} â€“ 6xy^{1+2}z

= – 4xy^{3}z^{2} â€“ 6xy^{3}z

Hence, the multiplication of 2xyz + 3xy and â€“ 2y^{2}z = – 4xy^{3}z^{2} â€“ 6xy^{3}z

(iv) â€“ 3xy^{2} + 4x^{2}y and â€“ xy

The multiplication of the given expression is calculated as,

(- 3xy^{2} + 4x^{2}y) Ã— – xy = 3x^{1+1}y^{2+1} â€“ 4x^{2+1}y^{1+1}

On calculation, we get

= 3x^{2}y^{3} â€“ 4x^{3}y^{2}

Hence, the multiplication of â€“ 3xy^{2} + 4x^{2}y and â€“ xy = 3x^{2}y^{3} â€“ 4x^{3}y^{2}

(v) 4xy and â€“ x^{2}y â€“ 3x^{2} y^{2}

The multiplication of the given expression is calculated as,

(- x^{2}y â€“ 3x^{2}y^{2}) Ã— 4xy = – x^{2}y Ã— 4xy â€“ 3x^{2}y^{2} Ã— 4xy

On further calculation, we get

= – 4x^{2+1}y^{1+1} â€“ 12x^{2+1}y^{2+1}

= – 4x^{3}y^{2} â€“ 12x^{3}y^{3}

Hence, the multiplication of 4xy and â€“ x^{2}y â€“ 3x^{2} y^{2} = – 4x^{3}y^{2} â€“ 12x^{3}y^{3}

**6. Multiply:**

**(i) 3a + 4b â€“ 5c and 3a**

**(ii) â€“ 5xy and â€“ xy ^{2} â€“ 6x^{2}y**

**Solution:**

(i) 3a + 4b â€“ 5c and 3a

The multiplication of the given expression is calculated as,

(3a + 4b â€“ 5c) Ã— 3a = 3a Ã— 3a + 4b Ã— 3a â€“ 5c Ã— 3a

On further calculation, we get

= 9a^{1+1} + 12ab â€“ 15ac

= 9a^{2} + 12ab â€“ 15ac

Therefore, the multiplication of 3a + 4b â€“ 5c and 3a = 9a^{2} + 12ab â€“ 15ac

(ii) â€“ 5xy and â€“ xy^{2} â€“ 6x^{2}y

The multiplication of the given expression is calculated as,

– 5xy Ã— (- xy^{2} â€“ 6x^{2}y) = – 5xy Ã— – xy^{2} â€“ 5xy Ã— – 6x^{2}y

On further calculation, we get

= 5x^{1+1}y^{1+2} + 30x^{1+2}y^{1+1}

= 5x^{2}y^{3} + 30x^{3}y^{2}

Therefore, the multiplication of â€“ 5xy and â€“ xy^{2} â€“ 6x^{2}y = 5x^{2}y^{3} + 30x^{3}y^{2}

**7. Multiply:**

**(i) x + 2 and x + 10**

**(ii) x + 5 and x â€“ 3**

**(iii) x â€“ 5 and x + 3**

**(iv) x â€“ 5 and x â€“ 3**

**(v) 2x + y and x + 3y**

**Solution:**

(i) x + 2 and x + 10

The given expression is calculated as follows

(x + 2) Ã— (x + 10) = x Ã— (x + 10) + 2 Ã— (x + 10)

We get,

= x^{2} + 10x + 2x + 20

= x^{2} + 12x + 20

Hence, the multiplication of (x + 2) and (x + 10) = x^{2} + 12x + 20

(ii) x + 5 and x â€“ 3

The given expression is calculated as follows

(x + 5) Ã— (x â€“ 3) = x Ã— (x â€“ 3) + 5 Ã— (x â€“ 3)

On simplification, we get

= x^{2} â€“ 3x + 5x â€“ 15

= x^{2} + 2x â€“ 15

Hence, the multiplication of (x + 5) and (x â€“ 3) = x^{2} + 2x â€“ 15

(iii) x â€“ 5 and x + 3

The given expression is calculated as follows

(x â€“ 5) Ã— (x + 3) = x Ã— (x + 3) â€“ 5 Ã— (x + 3)

On further calculation, we get

= x^{2} + 3x â€“ 5x â€“ 15

= x^{2} â€“ 2x – 15

Hence, the multiplication of (x â€“ 5) and (x + 3) = x^{2} â€“ 2x â€“ 15

(iv) x â€“ 5 and x â€“ 3

The given expression is calculated as,

(x â€“ 5) Ã— (x â€“ 3) = x Ã— (x â€“ 3) â€“ 5 Ã— (x â€“ 3)

On further calculation, we get

= x^{2} â€“ 3x â€“ 5x + 15

= x^{2} â€“ 8x + 15

Hence, the multiplication of (x â€“ 5) and (x â€“ 3) = x^{2} â€“ 8x + 15

(v) 2x + y and x + 3y

The given expression is calculated as,

(2x + y) Ã— (x + 3y) = 2x Ã— (x + 3y) + y Ã— (x + 3y)

On simplification, we get

= 2x^{2} + 6xy + xy + 3y^{2}

= 2x^{2} + 7xy + 3y^{2}

Hence, the multiplication of (2x + y) and (x + 3y) = 2x^{2} + 7xy + 3y^{2}

**8. Multiply:**

**(i) 3abc and â€“ 5a ^{2}b^{2}c**

**(ii) x â€“ y + z and -2x**

**(iii) 2x â€“ 3y â€“ 5z and -2y**

**(iv) â€“ 8xyz + 10 x ^{2}yz^{3} and xyz**

**(v) xyz and â€“ 13xy ^{2}z + 15x^{2}yz â€“ 6xyz^{2}**

**Solution:**

(i) 3abc and â€“ 5a^{2}b^{2}c

The given expression is calculated as follows,

3abc Ã— – 5a^{2}b^{2}c = 3 Ã— – 5 Ã— a Ã— a^{2} Ã— b Ã— b^{2} Ã— c Ã— c

On further calculation, we get

= – 15 Ã— a^{1+2} Ã— b^{1+2} Ã— c^{1+1}

= – 15 Ã— a^{3} Ã— b^{3} Ã— c^{2}

= – 15a^{3}b^{3}c^{2}

Therefore, the multiplication of 3abc and â€“ 5a^{2}b^{2}c = – 15a^{3}b^{3}c^{2}

(ii) x â€“ y + z and -2x

The given expression is calculated as follows,

(x â€“ y + z) Ã— – 2x = x Ã— – 2x â€“ y Ã— – 2x + z Ã— – 2x

On simplification, we get

= – 2x^{1+1} + 2xy â€“ 2xz

= – 2x^{2} + 2xy â€“ 2xz

Therefore, the multiplication of x â€“ y + z and -2x = – 2x^{2} + 2xy â€“ 2xz

(iii) 2x â€“ 3y â€“ 5z and -2y

The given expression is calculated as follows,

(2x â€“ 3y â€“ 5z) Ã— – 2y = 2x Ã— – 2y â€“ 3y Ã— – 2y â€“ 5z Ã— – 2y

On further calculation, we get

= – 4xy + 6y^{1+1} + 10yz

= – 4xy + 6y^{2} + 10yz

Therefore, the multiplication of 2x â€“ 3y â€“ 5z and -2y = – 4xy + 6y^{2} + 10yz

(iv) â€“ 8xyz + 10 x^{2}yz^{3} and xyz

The given expression is calculated as follows,

(- 8xyz + 10x^{2}yz^{3}) Ã— xyz = – 8xyz Ã— xyz + 10x^{2}yz^{3} Ã— xyz

On further calculation, we get

= – 8x^{1+1}y^{1+1}z^{1+1} + 10x^{2+1}y^{1+1}z^{3+1}

= – 8x^{2}y^{2}z^{2} + 10x^{3}y^{2}z^{4}

Therefore, the multiplication of â€“ 8xyz + 10 x^{2}yz^{3} and xyz = – 8x^{2}y^{2}z^{2} + 10x^{3}y^{2}z^{4}

(v) xyz and â€“ 13xy^{2}z + 15x^{2}yz â€“ 6xyz^{2}

The given expression is calculated as follows,

xyz Ã— (- 13xy^{2}z + 15x^{2}yz â€“ 6xyz^{2}) = xyz Ã— – 13xy^{2}z + xyz Ã—15x^{2}yz â€“ xyz Ã— 6xyz^{2}

On simplification, we get

= – 13x^{1+1}y^{1+2}z^{1+1} + 15x^{1+2}y^{1+1}z^{1+1} â€“ 6x^{1+1}y^{1+1}z^{1+2}

We get,

= – 13x^{2}y^{3}z^{2} + 15x^{3}y^{2}z^{2} â€“ 6x^{2}y^{2}z^{3}

Therefore, the multiplication of xyz and â€“ 13xy^{2}z + 15x^{2}yz â€“ 6xyz^{2} = – 13x^{2}y^{3}z^{2} + 15x^{3}y^{2}z^{2} â€“ 6x^{2}y^{2}z^{3}

**9. Find the product of:**

**(i) xy â€“ ab and xy + ab**

**(ii) 2abc â€“ 3xy and 2abc + 3xy**

**(iii) a + b â€“ c and 2a â€“ 3b**

**(iv) 5x â€“ 6y â€“ 7z and 2x + 3y**

**(v) 5x â€“ 6y â€“ 7z and 2x + 3y + z**

**Solution:**

(i) xy â€“ ab and xy + ab

The product of the given expression is calculated as,

(xy â€“ ab) Ã— (xy + ab) = xy Ã— (xy + ab) â€“ ab Ã— (xy + ab)

On simplification, we get

= xy Ã— xy + xy Ã— ab â€“ ab Ã— xy â€“ ab Ã— ab

= x^{2}y^{2} + abxy â€“ abxy â€“ a^{2}b^{2}

= x^{2}y^{2} â€“ a^{2}b^{2}

Hence, the product of (xy â€“ ab) and (xy + ab) = x^{2}y^{2} â€“ a^{2}b^{2}

(ii) 2abc â€“ 3xy and 2abc + 3xy

The product of the given expression is calculated as,

(2abc â€“ 3xy) Ã— (2abc + 3xy)

= 2abc Ã— (2abc + 3xy) â€“ 3xy Ã— (2abc + 3xy)

We get,

= 2abc Ã— 2abc + 2abc Ã— 3xy â€“ 3xy Ã— 2abc â€“ 3xy Ã— 3xy

= 4a^{2}b^{2}c^{2} + 6abcxy â€“ 6abcxy â€“ 9x^{2}y^{2}

= 4a^{2}b^{2}c^{2} â€“ 9x^{2}y^{2}

Hence, the product of 2abc â€“ 3xy and 2abc + 3xy = 4a^{2}b^{2}c^{2} â€“ 9x^{2}y^{2}

(iii) a + b â€“ c and 2a â€“ 3b

The product of the given expression is calculated as,

(a + b â€“ c) Ã— (2a â€“ 3b)

= a Ã— (2a â€“ 3b) + b Ã— (2a â€“ 3b) â€“ c Ã— (2a â€“ 3b)

= a Ã— 2a â€“ a Ã— 3b + b Ã— 2a â€“ b Ã— 3b â€“ c Ã— 2a + c Ã— 3b

= 2a^{1+1} â€“ 3ab + 2ab â€“ 3b^{1+1} â€“ 2ac + 3bc

We get,

= 2a^{2} â€“ ab â€“ 3b^{2} â€“ 2ac + 3bc

Hence, the product of a + b â€“ c and 2a â€“ 3b = 2a^{2} – ab â€“ 3b^{2} â€“ 2ac + 3bc

(iv) 5x â€“ 6y â€“ 7z and 2x + 3y

The product of the given expression is calculated as,

(5x â€“ 6y â€“ 7z) Ã— (2x + 3y)

= (5x â€“ 6y â€“ 7z) Ã— 2x + (5x â€“ 6y â€“ 7z) Ã— 3y

= 5x Ã— 2x â€“ 6y Ã— 2x â€“ 7z Ã— 2x + 5x Ã— 3y â€“ 6y Ã— 3y â€“ 7z Ã— 3y

We get,

= 10x^{2} â€“ 12xy â€“ 14xz + 15xy â€“ 18y^{2} â€“ 21yz

= 10x^{2} + 3xy â€“ 14xz â€“ 18y^{2} â€“ 21yz

Hence, the product of 5x â€“ 6y â€“ 7z and 2x + 3y = 10x^{2} + 3xy â€“ 14xz â€“ 18y^{2} â€“ 21yz

(v) 5x â€“ 6y â€“ 7z and 2x + 3y + z

The product of the given expression is calculated as,

(5x â€“ 6y â€“ 7z) Ã— (2x + 3y + z)

= (5x â€“ 6y â€“ 7z) Ã— 2x + (5x â€“ 6y â€“ 7z) Ã— 3y + (5x â€“ 6y â€“ 7z) Ã— z

= 5x Ã— 2x â€“ 6y Ã— 2x â€“ 7z Ã— 2x + 5x Ã— 3y â€“ 6y Ã— 3y â€“ 7z Ã— 3y + 5x Ã— z â€“ 6y Ã— z â€“ 7z Ã— z

We get,

= 10x^{2} â€“ 12xy â€“ 14xz + 15xy â€“ 18y^{2} â€“ 21yz + 5xz â€“ 6yz â€“ 7z^{2}

= 10x^{2} â€“ 12xy + 15xy â€“ 14xz + 5xz â€“ 18y^{2} â€“ 21yz â€“ 6yz â€“ 7z^{2}

= 10x^{2} + 3xy â€“ 9xz â€“ 18y^{2} â€“ 27yz â€“ 7z^{2}

Hence, the product of 5x â€“ 6y â€“ 7z and 2x + 3y + z = 10x^{2} + 3xy â€“ 9xz â€“ 18y^{2} â€“ 27yz â€“ 7z^{2}

Exercise 19(D)

**1. Divide:**

**(i) 3a by a**

**(ii) 15x by 3x**

**(iii) 16m by 4**

**(iv) 20x ^{2} by 5x**

**(v) 30p ^{2} by 10p^{2}**

**Solution:**

(i) 3a by a

3a Ã· a

This can be written as,

3a / a = (3 Ã— a) / a

= 3

Hence, 3a Ã· a = 3

(ii) 15x by 3x

15x Ã· 3x

15x / 3x = (15 Ã— x) / (3 Ã— x)

This can be written as,

= (3 Ã— 5 Ã— x) / (3 Ã— x)

We get,

= 5

Hence, 15x Ã· 3x = 5

(iii) 16m by 4

16m Ã· 4

16m / 4 = (16 Ã— m) / 4

This can be written as,

= (4 Ã— 4 Ã— m) / 4

We get,

= 4m

Hence, 16m Ã· 4 = 4m

(iv) 20x^{2} by 5x

20x^{2} Ã· 5x

20x^{2} / 5x = (20 Ã— x^{2}) / (5 Ã— x)

This can be written as,

= (4 Ã— 5 Ã— x^{2-1}) / 5

= 4 Ã— x

= 4x

Hence, 20x^{2} Ã· 5x = 4x

(v) 30p^{2} by 10p^{2}

30p^{2} Ã· 10p^{2} = (30 Ã— p^{2}) / (10 Ã— p^{2})

This can be written as,

= (3 Ã— 10 Ã— p^{2-2}) / 10

= 3 Ã— p^{0}

= 3 Ã— 1

= 3

Hence, 30p^{2} Ã· 10p^{2} = 3

**2. Simplify:**

**(i) 2x ^{5} Ã· x^{2}**

**(ii) 6a ^{8} Ã· 3a^{3}**

**(iii) 20xy Ã· – 5xy**

**(iv) â€“ 24a ^{2}b^{2}c^{2} Ã· 6ab**

**(v) â€“ 5x ^{2}y Ã· xy^{2}**

**Solution:**

(i) 2x^{5} Ã· x^{2}

= (2 Ã— x^{5}) / x^{2}

= 2 Ã— x^{5-2}

= 2 Ã— x^{3}

We get,

= 2x^{3}

Hence, 2x^{5} Ã· x^{2} = 2x^{3}

(ii) 6a^{8} Ã· 3a^{3}

= (6 Ã— a^{8}) / (3 Ã— a^{3})

This can be written as,

= (2 Ã— 3 Ã— a^{8-3}) / 3

We get,

= 2 Ã— a^{5}

= 2a^{5}

Hence, 6a^{8} Ã· 3a^{3} = 2a^{5}

(iii) 20xy Ã· – 5xy

= (20 Ã— x Ã— y) / (- 5 Ã— x Ã— y)

This can be written as,

= (4 Ã— 5) / – 5

We get,

= – 4

Hence, 20xy Ã· – 5xy = – 4

(iv) â€“ 24a^{2}b^{2}c^{2} Ã· 6ab

= (- 24 Ã— a^{2} Ã— b^{2} Ã— c^{2}) / (6 Ã— a Ã— b)

This can be written as,

= (-4 Ã— 6 Ã— a^{2-1} Ã— b^{2-1} Ã— c^{2}) / 6

We get,

= – 4 Ã— a Ã— b Ã— c^{2}

= – 4abc^{2}

Hence, â€“ 24a^{2}b^{2}c^{2} Ã· 6ab** = **– 4abc^{2}

(v) â€“ 5x^{2}y Ã· xy^{2}

= (- 5 Ã— x^{2} Ã— y) / (x Ã— y^{2})

This can be written as,

= (- 5 Ã— x^{2-1}) / y^{2-1}

We get,

= (- 5 Ã— x) / y

= – 5x / y

Hence, â€“ 5x^{2}y Ã· xy^{2} = – 5x / y

**3. Divide:**

**(i) (- 3m / 4) by 2m**

**(ii) â€“ 15p ^{6}q^{8} by – 5p^{6}q^{7}**

**(iii) â€“ 21m ^{5}n^{7} by 14m^{2}n^{2}**

**(iv) 36a ^{4}x^{5}y^{6} by 4x^{2}a^{3}y^{2}**

**(v) 20x ^{3}a^{6} by 5xy**

**Solution:**

(i) (- 3m / 4) by 2m

= – 3m / 4 Ã· 2m = – 3m / 4 Ã— 1 / 2m

= – (3 Ã— m) / (4 Ã— 2 Ã— m)

We get,

= – 3 / 8

Hence, (- 3m / 4) Ã· 2m = – 3 / 8

(ii) â€“ 15p^{6}q^{8} by – 5p^{6}q^{7}

– 15p^{6}q^{8} Ã· – 5p^{6}q^{7} = (- 15 Ã— p^{6} Ã— q^{8}) / (- 5 Ã— p^{6} Ã— q^{7})

This can be written as,

= (3 Ã— 5 Ã— q^{8-7}) / 5

We get,

= 3 Ã— q

= 3q

Hence, – 15p^{6}q^{8} Ã· – 5p^{6}q^{7} = 3q

(iii) â€“ 21m^{5}n^{7} by 14m^{2}n^{2}

– 21m^{5}n^{7} Ã· 14m^{2}n^{2} = (- 21 Ã— m^{5} Ã— n^{7}) / (14 Ã— m^{2} Ã— n^{2})

This can be written as,

= (- 3 Ã— 7 Ã— m^{5-2} Ã— n^{7-2}) / (2 Ã— 7)

= (- 3 Ã— m^{3} Ã— n^{5}) / 2

We get,

= – 3m^{3}n^{5} / 2

Hence, – 21m^{5}n^{7} Ã· 14m^{2}n^{2} = – 3m^{3}n^{5} / 2

(iv) 36a^{4}x^{5}y^{6} by 4x^{2}a^{3}y^{2}

36a^{4}x^{5}y^{6} Ã· 4x^{2}a^{3}y^{2} = (36 Ã— a^{4} Ã— x^{5} Ã— y^{6}) / (4 Ã— x^{2} Ã— a^{3} Ã— y^{2})

This can be written as,

= (4 Ã— 9 Ã— a^{4-3} Ã— x^{5-2} Ã— y^{6-2}) / 4

= 9 Ã— a^{1} Ã— x^{3} Ã— y^{4}

We get,

= 9ax^{3}y^{4}

Hence, 36a^{4}x^{5}y^{6} Ã· 4x^{2}a^{3}y^{2} = 9ax^{3}y^{4}

(v) 20x^{3}a^{6} by 5xy

20x^{3}a^{6} Ã· 5xy = (20 Ã— x^{3} Ã— a^{6}) / (5 Ã— x Ã— y)

This can be written as,

= (4 Ã— 5 Ã— x^{3-1} Ã— a^{6}) / (5 Ã— y)

We get,

= (4 Ã— x^{2} Ã— a^{6}) / y

= 4x^{2}a^{6} / y

Hence, 20x^{3}a^{6} Ã· 5xy = 4x^{2}a^{6} / y

**4. Simplify:**

**(i) (- 15m ^{5}n^{2}) / (- 3m^{5})**

**(ii) 35x ^{4}y^{2} / – 15x^{2}y^{2}**

**(iii) (- 24x ^{6}y^{2}) / (6x^{6}y)**

**Solution:**

(i) (-15m^{5}n^{2}) / (- 3m^{5}) = (-15 Ã— m^{5} Ã— n^{2}) / (- 3 Ã— m^{5})

This can be written as,

= (3 Ã— 5 Ã— m^{5-5} Ã— n^{2}) / 3

= 5 Ã— m^{0} Ã— n^{2}

= 5 Ã— 1 Ã— n^{2}

= 5n^{2}

Hence, (-15m^{5}n^{2}) / (- 3m^{5}) = 5n^{2}

(ii) 35x^{4}y^{2} / – 15x^{2}y^{2}

35x^{4}y^{2} / – 15x^{2}y^{2} = (35 Ã— x^{4} Ã— y^{2}) / (- 15 Ã— x^{2} Ã— y^{2})

This can be written as,

= – (5 Ã— 7 Ã— x^{4-2} Ã— y^{2-2}) / (3 Ã— 5)

= – (7 Ã— x^{2} Ã— y^{0}) / 3

We get,

= – 7x^{2}y / 3

Hence, 35x^{4}y^{2} / – 15x^{2}y^{2} = – 7x^{2}y / 3

(iii) (- 24x^{6}y^{2}) / (6x^{6}y)

(- 24x^{6}y^{2}) / (6x^{6}y) = (- 24 Ã— x^{6} Ã— y^{2}) / (6 Ã— x^{6} Ã— y)

This can be written as,

= (- 4 Ã— 6 Ã— x^{6-6} Ã— y^{2-1}) / 6

= – 4 Ã— x^{0} Ã— y^{1}

= – 4y

Hence, (- 24x^{6}y^{2}) / (6x^{6}y) = – 4y

**5. Divide:**

**(i) 9x ^{3} â€“ 6x^{2} by 3x**

**(ii) 6m ^{2} â€“ 16m^{3} + 10m^{4} by â€“ 2m**

**(iii) 15x ^{3}y^{2} + 25x^{2}y^{3} â€“ 36x^{4}y^{4} by 5x^{2}y^{2}**

**(iv) 36a ^{3}x^{5} â€“ 24a^{4}x^{4} + 18a^{5}x^{3} by â€“ 6a^{3}x^{3}**

**Solution:**

(i) 9x^{3} â€“ 6x^{2} by 3x

9x^{3} â€“ 6x^{2} Ã· 3x = (9 Ã— x^{3} â€“ 6 Ã— x^{2}) / (3 Ã— x)

Separating the terms, we get

= (9 Ã— x^{3}) / (3 Ã— x) â€“ (6 Ã— x^{2}) / (3 Ã— x)

We get,

= 3 Ã— x^{3-1} â€“ 2 Ã— x^{2-1}

= 3x^{2} â€“ 2x

Hence, 9x^{3} â€“ 6x^{2} Ã· 3x = 3x^{2} â€“ 2x

(ii) 6m^{2} â€“ 16m^{3} + 10m^{4} by â€“ 2m

6m^{2 }â€“ 16m^{3} + 10m^{4} Ã· – 2m = (6 Ã— m^{2}– 16 Ã— m^{3} + 10 Ã— m^{4}) / – 2 Ã— m

Separating the terms, we get

= (6 Ã— m^{2} / – 2 Ã— m) â€“ (16 Ã— m^{3}) / (- 2 Ã— m) + (10 Ã— m^{4}) / (- 2 Ã— m)

= – 3 Ã— m^{2-1} + 8 Ã— m^{3-1} â€“ 5 Ã— m^{4-1}

= – 3 Ã— m + 8 Ã— m^{2} â€“ 5 Ã— m^{3}

We get,

= – 3m + 8m^{2} â€“ 5m^{3}

Hence, 6m^{2 }â€“ 16m^{3} + 10m^{4} Ã· – 2m = – 3m + 8m^{2} â€“ 5m^{3}

(iii) 15x^{3}y^{2} + 25x^{2}y^{3} â€“ 36x^{4}y^{4} by 5x^{2}y^{2}

15x^{3}y^{2} + 25x^{2}y^{3} â€“ 36x^{4}y^{4} Ã· 5x^{2}y^{2} = (15x^{3}y^{2} + 25x^{2}y^{3} â€“ 36x^{4}y^{4}) / (5x^{2}y^{2})

= (15 Ã— x^{3} Ã— y^{2}) / (5 Ã— x^{2} Ã— y^{2}) + (25 Ã— x^{2} Ã— y^{3}) / (5 Ã— x^{2} Ã— y^{2}) â€“ (36 Ã— x^{4} Ã— y^{4}) / (5 Ã— x^{2} Ã— y^{2})

On further calculation, we get

= 3 Ã— x^{3-2} Ã— y^{2-2} + 5 Ã— x^{2-2} Ã— y^{3-2} â€“ (36 Ã— x^{4-2} Ã— y^{4-2}) / 5

We get,

= 3 Ã— x^{1} Ã— y^{0} + 5 Ã— x^{0} Ã— y^{1} â€“ (36 Ã— x^{2} Ã— y^{2}) / 5

= 3x + 5y â€“ (36x^{2}y^{2}) / 5

Hence, 15x^{3}y^{2} + 25x^{2}y^{3} â€“ 36x^{4}y^{4} Ã· 5x^{2}y^{2} = 3x + 5y â€“ (36x^{2}y^{2}) / 5

(iv) 36a^{3}x^{5} â€“ 24a^{4}x^{4} + 18a^{5}x^{3} by â€“ 6a^{3}x^{3}

36a^{3}x^{5} â€“ 24a^{4}x^{4} + 18a^{5}x^{3} Ã· (â€“ 6a^{3}x^{3}) = (36a^{3}x^{5} â€“ 24a^{4}x^{4} + 18a^{5}x^{3}) / – 6a^{3}x^{3}

= (36.a^{3}.x^{5}) / (-6.a^{3}.x^{3}) â€“ (24.a^{4}.x^{4}) / (-6.a^{3}.x^{3}) + (18.a^{5}.x^{3}) / (-6.a^{3}.x^{3})

We get,

= – 6.x^{5-3} + 4.a^{4-3}.x^{4-3} â€“ 3.a^{5-3}

= – 6x^{2} + 4ax â€“ 3a^{2}

Hence, 36a^{3}x^{5} â€“ 24a^{4}x^{4} + 18a^{5}x^{3} Ã· (â€“ 6a^{3}x^{3}) = – 6x^{2} + 4ax â€“ 3a^{2}