Selina Solutions Concise Mathematics Class 6 Chapter 19: Fundamental Operations

Selina Solutions Concise Mathematics Class 6 Chapter 19 Fundamental Operations has accurate answers, designed by subject matter experts at BYJU’S, in accordance with the grasping abilities of students. The solutions play a key role in improving students’ 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.

Selina Solutions Concise Mathematics Class 6 Chapter 19: Fundamental Operations Download PDF

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

Exercise 19(A) Solutions

Exercise 19(B) Solutions

Exercise 19(C) Solutions

Exercise 19(D) Solutions

Access Selina Solutions Concise Mathematics Class 6 Chapter 19: Fundamental Operations

Exercise 19(A)

1. Fill in the blanks:

(i) 5 + 4 = …….. and 5x + 4x = ………

(ii) 12 + 18 = ……. and 12x²y + 18x²y = ……..

(iii) 7 + 16 = …….. and 7a + 16b = …….

(iv) 1 + 3 = ……. and x²y + 3xy² = ……

(v) 7 – 4 = …… and 7ab – 4ab = ………

Solution:

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

(ii) 12 + 18 = 30 and 12x²y + 18x²y = 30x²y

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

(iv) 1 + 3 = 4 and x²y + 3xy² = x²y + 3xy²

(v) 7 – 4 = 3 and 7ab – 4ab = 3ab

2. Fill in the blanks:

(i) The sum of -2 and -5 = ……… and the sum of -2x and -5x = ……….

(ii) The sum of 8 and -3 = ……… and the sum of 8ab and -3ab = ………….

(iii) The sum of -15 and -4 = ……… and the sum of -15x and -4y = ………..

(iv) 15 + 8 + 3 = …….. and 15x + 8y + 3x = …………

(v) 12 – 9 + 15 = ………… and 12ab – 9ab + 15ba = ……….

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 – 4y

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

(v) 12 – 9 + 15 = 18 and 12ab – 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 – a – 5a – 2a

(ii) 2b – 3b – b + 4b

(iii) 3x – 2x – 4x + 7x

(iv) 5ab + 2ab – 6ab + ab

(v) 8x – 5y – 3x + 10y

Solution:

(i) 6a – a – 5a – 2a

The value of given expression is calculated as below

6a – a – 5a – 2a = (6 – 1 – 5 – 2) a

We get,

= (5 – 5 – 2) a

= – 2a

Therefore, 6a – a – 5a – 2a = – 2a

(ii) 2b – 3b – b + 4b

The value of given expression is calculated as below

2b – 3b – b + 4b = 2b + 4b – (3 + 1) b

We get,

= 6b – 4b

= 2b

Therefore, 2b – 3b – b + 4b = 2b

(iii) 3x – 2x – 4x + 7x

The given expression is calculated as below

3x – 2x – 4x + 7x = 3x + 7x – 2x – 4x

= (3 + 7) x – (2 + 4) x

= 10x – 6x

= 4x

Therefore, 3x – 2x – 4x + 7x = 4x

(iv) 5ab + 2ab – 6ab + ab

The given expression is calculated as below

5ab + 2ab – 6ab + ab = 5ab + 2ab + ab – 6ab

We get,

= 8ab – 6ab

= 2ab

Therefore, 5ab + 2ab – 6ab + ab = 2ab

(v) 8x – 5y – 3x + 10y

The given expression is calculated as below

8x – 5y – 3x + 10y = 8x – 3x + 10y – 5y

= 5x + 5y

Therefore, 8x – 5y – 3x + 10y = 5x + 5y

5. Evaluate:

(i) -7x + 9x + 2x – 2x

(ii) 5ab – 2ab – 8ab + 6ab

(iii) -8a – 3a + 12a + 13a – 6a

(iv) 19abc – 11abc – 12abc + 14abc

Solution:

(i) -7x + 9x + 2x – 2x

The value of given expression is calculated as follows

-7x + 9x + 2x – 2x = 9x + 2x – 7x – 2x

= 11x – 9x

We get,

= 2x

Hence, -7x + 9x + 2x – 2x = 2x

(ii) 5ab – 2ab – 8ab + 6ab

The value of given expression is calculated as follows

5ab – 2ab – 8ab + 6ab = 5ab + 6ab – 2ab – 8ab

We get,

= 11ab – 10ab

= ab

Hence, 5ab – 2ab – 8ab + 6ab = ab

(iii) –8a – 3a + 12a + 13a – 6a

The value of given expression is calculated as follows

-8a – 3a + 12a + 13a – 6a = 12a + 13a – (8a + 3a + 6a)

= 25a – 17a

= 8a

Hence, -8a – 3a + 12a + 13a – 6a = 8a

(iv) 19abc – 11abc – 12abc + 14abc

The value of given expression is calculated as follows

19abc – 11abc – 12abc + 14abc = abc (19 – 11 – 12 + 14)

= abc (33 – 23)

= 10abc

Hence, 19abc – 11abc – 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 – 4ab = 2ab

(ii) 4.8b, 6.8b

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

6.8b – 4.8b = 2b

(iii) 3.5abc, 10.5abc

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

10.5abc – 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 – 3 (1 / 2) mn = (17 / 2)nm – (7 / 2)mn

We get,

= [(17mn – 7mn) / 2]

= (10 / 2)mn

= 5mn

7. Simplify:

(i) 2a²b² + 5ab² + 8a²b² – 3ab²

(ii) 4a + 3b – 2a – b

(iii) 2xy + 4yz + 5xy + 3yz – 6xy

(iv) ab + 15ab – 11ab – 2ab

(v) 6a² – 3b² + 2a² + 5b² – 4a²

Solution:

(i) 2a²b² + 5ab² + 8a²b² – 3ab²

The simplified form of the given expression is calculated as follows

2a²b² + 5ab² + 8a²b² – 3ab² = 2a²b² + 8a²b² + 5ab² – 3ab²

We get,

= 10a²b² + 2ab²

Therefore, 2a²b² + 5ab² + 8a²b² – 3ab² = 10a²b² + 2ab²

(ii) 4a + 3b – 2a – b

The simplified form of the given expression is calculated as follows

4a + 3b – 2a – b = 4a – 2a + 3b – b

= 2a + 2b

Therefore, 4a + 3b – 2a – b = 2a + 2b

(iii) 2xy + 4yz + 5xy + 3yz – 6xy

The simplified form of the given expression is calculated as follows

2xy + 4yz + 5xy + 3yz – 6xy = 2xy + 5xy – 6xy + 4yz + 3yz

= xy + 7yz

Therefore, 2xy + 4yz + 5xy + 3yz – 6xy = xy + 7yz

(iv) ab + 15ab – 11ab – 2ab

The simplified form of the given expression is calculated as follows

ab + 15ab – 11ab – 2ab = 16ab – 13ab

= 3ab

Therefore, ab + 15ab – 11ab – 2ab = 3ab

(v) 6a² – 3b² + 2a² + 5b² – 4a²

The simplified form of the given expression is calculated as follows

6a² – 3b² + 2a² + 5b² – 4a² = 6a² + 2a² – 4a² + 5b² – 3b²

We get,

= 4a² + 2b²

Therefore, 6a² – 3b² + 2a² + 5b² – 4a² = 4a² + 2b²

Exercise 19(B)

1. Find the sum of:

(i) 3a + 4b + 7c, – 5a + 3b – 6c

and 4a – 2b – 4c

(ii) 2x² + xy – y², – x² + 2xy + 3y²

and 3x² – 10xy + 4y²

(iii) x² – x + 1, -5x² + 2x – 2

and 3x² – 3x + 1

(iv) a² – ab + bc, 2ab + bc – 2a²

and – 3bc + 3a² + ab

(v) 4x² + 7 – 3x, 4x – x² + 8

and – 10 + 5x – 2x²

Solution:

(i) 3a + 4b + 7c, – 5a + 3b – 6c

and 4a – 2b – 4c

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

(3a + 4b + 7c) + (-5a + 3b – 6c) + (4a – 2b – 4c)

= 3a + 4a – 5a + 4b + 3b – 2b + 7c – 6c – 4c

We get,

= 2a +5b – 3c

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

(ii) 2x² + xy – y², – x² + 2xy + 3y²

and 3x² – 10xy + 4y²

The sum of 2x² + xy – y², – x² + 2xy + 3y² and 3x² – 10xy + 4y² is calculated as shown below

(2x² + xy – y²) + (- x² + 2xy + 3y²) + (3x² – 10xy + 4y²)

= 2x² – x² + 3x² + xy + 2xy – 10xy + 3y² + 4y² – y²

We get,

= 4x² – 7xy + 6y²

Hence, the sum of 2x² + xy – y², – x² + 2xy + 3y² and 3x² – 10xy + 4y² is 4x² – 7xy + 6y²

(iii) x² – x + 1, -5x² + 2x – 2 and 3x² – 3x + 1

The sum of (x² – x + 1), (- 5x² + 2x – 2) and (3x² – 3x + 1) is calculated as shown below

(x² – x + 1) + (- 5x² + 2x – 2) + (3x² – 3x + 1)

= x² – 5x² + 3x² + 2x – x – 3x + 1 +1 – 2

= – x² – 2x

Hence, the sum of (x² – x + 1), (- 5x² + 2x – 2) and (3x² – 3x + 1) is – x² – 2x

(iv) a² – ab + bc, 2ab + bc – 2a² and – 3bc + 3a² + ab

The sum of (a² – ab + bc), (2ab + bc – 2a²) and (- 3bc + 3a² + ab) is calculated as shown below

(a² – ab + bc) + (2ab + bc – 2a²) + (- 3bc + 3a² + ab)

= a² – 2a² + 3a² + 2ab + ab – ab + bc + bc – 3bc

We get,

= 2a² +2ab – bc

Hence, the sum of (a² – ab + bc), (2ab + bc – 2a²) and (- 3bc + 3a² + ab) is 2a² +2ab – bc

(v) 4x² + 7 – 3x, 4x – x² + 8 and – 10 + 5x – 2x²

The sum of (4x² + 7 – 3x), (4x – x² + 8) and (- 10 + 5x – 2x²) is calculated as shown below

(4x² + 7 – 3x) + (4x – x² + 8) + (- 10 + 5x – 2x²)

= 4x² – x² – 2x² + 7 + 8 – 10 + 4x + 5x – 3x

We get,

= x² + 5 + 6x

Hence, the sum of (4x² + 7 – 3x), (4x – x² + 8) and (- 10 + 5x – 2x²) is x² + 5 + 6x

2.Add the following expressions:

(i) – 17x² – 2xy + 23y², – 9y² + 15x² + 7xy

and 13x² + 3y² – 4xy

(ii) – x² – 3xy + 3y² + 8, 3x² – 5y² – 3 + 4xy

and – 6xy + 2x² – 2 + y²

(iii) a³ – 2b³ + a, b³ – 2a³ + b

and – 2b + 2b³ – 5a + 4a³

Solution:

(i) The sum of (– 17x² – 2xy + 23y²), (- 9y² + 15x² + 7xy) and (13x² + 3y² – 4xy) is calculated as follows

(– 17x² – 2xy + 23y²) + (- 9y² + 15x² + 7xy) + (13x² + 3y² – 4xy)

= – 17x² + 15x² + 13x² – 2xy – 4xy + 7xy + 23y² + 3y² – 9y²

We get,

= 11x² + xy + 17y²

Therefore, the sum of (– 17x² – 2xy + 23y²), (- 9y² + 15x² + 7xy) and (13x² + 3y² – 4xy) is 11x² + xy + 17y²

(ii) – x² – 3xy + 3y² + 8, 3x² – 5y² – 3 + 4xy and – 6xy + 2x² – 2 + y²

The sum of (– x² – 3xy + 3y² + 8), (3x² – 5y² – 3 + 4xy) and (– 6xy + 2x² – 2 + y²) is calculated as follows

(– x² – 3xy + 3y² + 8) + (3x² – 5y² – 3 + 4xy) + (– 6xy + 2x² – 2 + y²)

= – x² + 3x² + 2x² – 3xy – 6xy + 4xy + 3y² + y² – 5y² + 8 – 3 – 2

We get,

= 4x² – 5xy – y² + 3

Therefore, the sum of (– x² – 3xy + 3y² + 8), (3x² – 5y² – 3 + 4xy) and (– 6xy + 2x² – 2 + y²) is 4x² – 5xy – y² + 3

(iii) a³ – 2b³ + a, b³ – 2a³ + b and – 2b + 2b³ – 5a + 4a³

The sum of (a³ – 2b³ + a), (b³ – 2a³ + b) and (– 2b + 2b³ – 5a + 4a³) is calculated as follows

(a³ – 2b³ + a) + (b³ – 2a³ + b) + (– 2b + 2b³ – 5a + 4a³)

= a³ + 4a³ – 2a³ – 2b³ + b³ + 2b³ + a – 5a + b – 2b

We get,

= 3a³ + b³ – 4a – b

Therefore, the sum of (a³ – 2b³ + a), (b³ – 2a³ + b) and (– 2b + 2b³ – 5a + 4a³) is 3a³ + b³ – 4a – b

3. Evaluate:

(i) 3a – (a + 2b)

(ii) (5x – 3y) – (x + y)

(iii) (8a + 15b) – (3b – 7a)

(iv) (8x + 7y) – (4y – 3x)

(v) 7 – (4a – 5)

Solution:

(i) 3a – (a + 2b)

The value of the given expression is calculated as below

3a – (a + 2b)

= 3a – a – 2b

= 2a – 2b

Taking 2 as common, we get

= 2(a – b)

(ii) (5x – 3y) – (x + y)

The value of the given expression is calculated as below

(5x – 3y) – (x + y)

= 5x – x – 3y – y

= 4x – 4y

Taking 4 as common, we get

= 4(x – y)

(iii) (8a + 15b) – (3b – 7a)

The value of the given expression is calculated as below

(8a + 15b) – (3b – 7a)

= 8a + 7a + 15b – 3b

On calculation, we get

= 15a + 12b

(iv) (8x + 7y) – (4y – 3x)

The value of the given expression is calculated as below

(8x + 7y) – (4y – 3x)

= 8x + 3x + 7y – 4y

On further calculation, we get

= 11x + 3y

(v) 7 – (4a – 5)

The value of the given expression is calculated as below

7 – (4a – 5)

= 7 – 4a + 5

We get,

= 12 – 4a

4. Subtract:

(i) 5a – 3b + 2c from a – 4b – 2c

(ii) 4x – 6y + 3z from 12x + 7y – 21z

(iii) 5 – a – 4b + 4c from 5a – 7b + 2c

(iv) – 8x – 12y + 17z from x – y – z

(v) 2ab + cd – ac – 2bd from ab – 2cd + 2ac + bd

Solution:

(i) 5a – 3b + 2c from a – 4b – 2c

The value of the subtraction is calculated as follows

(a – 4b – 2c) – (5a – 3b + 2c)

= a – 5a – 4b + 3b – 2c – 2c

We get,

= – 4a – b – 4c

(ii) 4x – 6y + 3z from 12x + 7y – 21z

The value of the subtraction is calculated as follows

(12x + 7y – 21z) – (4x – 6y + 3z)

= 12x – 4x + 7y + 6y – 21z – 3z

On further calculation, we get

= 8x + 13y – 24z

(iii) 5 – a – 4b + 4c from 5a – 7b + 2c

The value of the subtraction is calculated as follows

(5a – 7b + 2c) – (5 – a – 4b + 4c)

= 5a + a – 7b + 4b + 2c – 4c – 5

We get,

=6a – 3b – 2c – 5

(iv) – 8x – 12y + 17z from x – y – z

The value of the subtraction is calculated as follows

(x – y – z) – (- 8x – 12y + 17z)

= x + 8x + 12y – y – z – 17z

We get,

= 9x + 11y – 18z

(v) 2ab + cd – ac – 2bd from ab – 2cd + 2ac + bd

The value of the subtraction is calculated as follows

(ab – 2cd + 2ac + bd) – (2ab + cd – ac – 2bd)

= ab – 2ab – 2cd – cd + 2ac + ac + bd + 2bd

On calculating further, we get

= – ab – 3cd + 3ac + 3bd

5.

(i) Take – ab + bc – ca from bc – ca + ab.

(ii) Take 5x + 6y – 3z from 3x + 5y – 4z.

(iii) Take (-3 / 2) p + q – r from (1 / 2)p – (1 / 3)q – (3 / 2) r

(iv) Take 1 – a + a² from a² + a + 1

Solution:

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

(bc – ca + ab) – (-ab + bc – ca)

= bc – bc – ca + ca + ab + ab

We get,

= 2ab

Hence, (bc – ca + ab) – (-ab + bc – ca) = 2ab

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

(3x + 5y – 4z) – (5x + 6y – 3z)

= 3x – 5x + 5y – 6y – 4z + 3z

On simplification, we get

= – 2x – y – z

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

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

= (1 / 2)p + (3 / 2)p – (1 / 3)q – q – (3 / 2)r + r

On further calculation, we get

= 2p – (4 / 3)q – (1 / 2)r

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

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

(a² + a + 1) – (1 – a + a²)

= a² – a² + a + a + 1 – 1

We get,

= a + a

= 2a

6. From the sum of x + y – 2z and 2x – y + z subtract x + y + z.

Solution:

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

(x + y – 2z) + (2x – y + z) – (x + y + z)

= x + 2x – x + y – y – y – 2z – z + z

We get,

= 2x – y – 2z

Therefore, (x + y – 2z) + (2x – y + z) – (x + y + z) = 2x – y – 2z

7. From the sum of 3a – 2b + 4c and 3b – 2c subtract a – b – c.

Solution:

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

(3a – 2b + 4c) + (3b – 2c) – (a – b – c)

= 3a – 2b + 4c + 3b – 2c – a + b + c

On further calculation, we get

= 3a – a + 3b + b – 2b + 4c + c – 2c

= 2a + 2b + 3c

Hence, (3a – 2b + 4c) + (3b – 2c) – (a – b – c) = 2a + 2b + 3c

8. Subtract x – 2y – z from the sum of 3x – y + z and x + y – 3z.

Solution:

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

(3x – y + z) + (x + y – 3z) – (x – 2y – z)

= 3x + x – x – y + y + 2y + z – 3z + z

We get,

= 3x + 2y – z

Therefore, (3x – y + z) + (x + y – 3z) – (x – 2y – z) = 3x + 2y – z

9. Subtract the sum of x + y and x – z from the sum of x – 2z and x + y + z

Solution:

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

(x – 2z) + (x + y + z) – {(x + y) + (x – z)}

On further calculation, we get

= x + x – x –x + y – y + z + z – 2z

We get,

= 0

Therefore, (x – 2z) + (x + y + z) – {(x + y) + (x – z)} = 0

10. By how much should x + 2y – 3z be increased to get 3x?

Solution:

The terms calculated as per the question is as follows

3x – (x + 2y – 3z)

= 3x – x – 2y + 3z

We get,

= 2x – 2y + 3z

11. The sum of two expressions is 5x² – 3y². If one of them is 3x² + 4xy – y², find the other.

Solution:

The other expression is calculated as follows

(5x² – 3y²) – (3x² + 4xy – y²)

= 5x² – 3x² – 4xy – 3y² + y²

We get,

= 2x² – 4xy – 2y²

12. The sum of two expressions is 3a² + 2ab – b². If one of them is 2a² + 3b², find the other.

Solution:

The other expression is calculated as follows

(3a² + 2ab – b²) – (2a² + 3b²)

On simplification, we get

=3a² – 2a² – b² – 3b² + 2ab

= a²– 4b² + 2ab

Exercise 19(C)

1. Fill in the blanks:

(i) 6 × 3 = ………. and 6x × 3x = …………

(ii) 6 × 3 = ……… and 6x² × 3x³ = …………

(iii) 5 × 4 = ………. and 5x × 4y = …………

(iv) 4 × 7 = …………. and 4ax × 7x = ……….

(v) 6 × 2 = …………. and 6xy × 2xy = ……….

Solution:

(i) 6 × 3 = 18

Hence,

6x × 3x = 6 × 3 × x × x

We get,

= 18 × x²

= 18x²

Therefore, 6 × 3 = 18 and 6x × 3x = 18x²

(ii) 6 × 3 = 18

Hence,

6x² × 3x³ = 6 × 3 × x^{2 + 3}

= 18 × x⁵

= 18x⁵

Therefore, 6 × 3 = 18 and 6x² × 3x³ = 18x⁵

(iii) 5 × 4 = 20 and 5x × 4y = 5 × 4 × x × y

= 20xy

Therefore, 5 × 4 = 20 and 5x × 4y = 20xy

(iv) 4 × 7 = 28

Hence,

4ax × 7x = 4 × 7 × a × x × x

= 28 × a × x²

= 28ax²

Therefore, 4 × 7 = 28 and 4ax × 7x = 28ax²

(v) 6 × 2 = 12

Hence,

6xy × 2xy = 6 × 2 × x^{1 + 1} × y^{1 + 1}

= 12 × x² × y²

= 12x²y²

Therefore, 6 × 2 = 12 and 6xy × 2xy = 12x²y²

2. Fill in the blanks:

(i) 4x × 6x × 2 = ………

(ii) 3ab × 6ax = …………

(iii) x × 2x² × 3x³ = ………

(iv) 5 × 5a³ = …………

(v) 6 × 6x² × 6x²y² = ………

Solution:

(i) 4x × 6x × 2 = 4 × 6 × 2 × x × x

= 48 × x²

We get,

= 48x²

Hence, 4x × 6x × 2 = 48x²

(ii) 3ab × 6ax = 3 × 6 × a × a × b × x

= 18 × a² × b × x

We get,

= 18a²bx

Hence, 3ab × 6ax = 18a²bx

(iii) x × 2x² × 3x³ = 2 × 3 × x × x² × x³

= 6 × x^{1 + 2 + 3}

= 6 × x⁶

= 6x⁶

Hence, x × 2x² × 3x³ = 6x⁶

(iv) 5 × 5a³ = 5 × 5 × a³

= 25 × a³

We get,

= 25a³

Hence, 5 × 5a³ = 25a³

(v) 6 × 6x² × 6x²y² = 6 × 6 × 6 × x² × x² × y²

= 216 × x^{2+ 2} × y²

= 216 × x⁴ × y²

We get,

= 216x⁴y²

Hence, 6 × 6x² × 6x²y² = 216x⁴y²

3. Find the value of:

(i) 3x³ × 5x⁴

(ii) 5a² × 7a⁷

(iii) 3abc × 6ac³

(iv) a²b² × 5a³b⁴

(v) 2x²y³ × 5x³y⁴

Solution:

(i) 3x³ × 5x⁴

3x³ × 5x⁴ = 3 × 5 × x³ × x⁴

= 15 × x^{3 + 4}

We get,

= 15 × x⁷

= 15x⁷

Hence, the value of 3x³ × 5x⁴ is 15x⁷

(ii) 5a² × 7a⁷

5a² × 7a⁷ = 5 × 7 × a² × a⁷

= 35 × a^{2 + 7}

= 35 × a⁹

We get,

= 35a⁹

Hence, the value of 5a² × 7a⁷ is 35a⁹

(iii) 3abc × 6ac³

3abc × 6ac³ = 3 × 6 × a × a × b × c × c³

=18 × a^{1+ 1} × b × c¹⁺³

= 18 × a² × b × c⁴

We get,

= 18a²bc⁴

Hence, the value of 3abc × 6ac³ is 18a²bc⁴

(iv) a²b² × 5a³b⁴

a²b² × 5a³b⁴ = 5 × a² × a³ × b² × b⁴

= 5 × a²⁺³ × b²⁺⁴

= 5 × a⁵ × b⁶

We get,

= 5a⁵b⁶

Hence, the value of a²b² × 5a³b⁴ is 5a⁵b⁶

(v) 2x²y³ × 5x³y⁴

2x²y³ × 5x³y⁴ = 2 × 5 × x² × x³ × y³ × y⁴

= 10 × x²⁺³ × y³⁺⁴

We get,

= 10 × x⁵ × y⁷

= 10x⁵y⁷

Hence, the value of 2x²y³ × 5x³y⁴ is 10x⁵y⁷

4. Multiply:

(i) a + b by ab

(ii) 3ab – 4b by 3ab

(iii) 2xy – 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) × ab = a × ab + b × ab

= a¹⁺¹b + ab¹⁺¹

We get,

= a²b + ab²

Hence, (a + b) by ab = a²b + ab²

(ii) 3ab – 4b by 3ab

The multiplication of 3ab – 4b by 3ab is calculated as,

(3ab – 4b) × 3ab = 3ab × 3ab – 4b × 3ab

= 9a¹⁺¹b¹⁺¹ – 12ab¹⁺¹

We get,

= 9a²b² – 12ab²

Therefore, (3ab – 4b) by 3ab = 9a²b² – 12ab²

(iii) 2xy – 5by by 4bx

The multiplication of 2xy – 5by by 4bx is calculated as,

(2xy – 5by) × 4bx = 2xy × 4bx – 5by × 4bx

= 8bx¹⁺¹y – 20b¹⁺¹xy

We get,

= 8bx²y – 20b²xy

Therefore, (2xy – 5by) by 4bx = 8bx²y – 20b²xy

(iv) 4x + 2y by 3xy

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

(4x + 2y) × 3xy = 4x × 3xy + 2y × 3xy

On simplification, we get

= 12x¹⁺¹y + 6xy¹⁺¹

= 12x²y + 6xy²

Therefore, (4x + 2y) by 3xy = 12x²y + 6xy²

(v) 1 + 4x by x

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

(1 + 4x) × x = 1 × x + 4x × x

On simplification, we get

= x + 4x¹⁺¹

= x + 4x²

Therefore, (1 + 4x) by x = x + 4x²

5. Multiply:

(i) – x + y – z and – 2x

(ii) xy – yz and x²yz²

(iii) 2xyz + 3xy and – 2y²z

(iv) – 3xy² + 4x²y and – xy

(v) 4xy and – x²y – 3x² y²

Solution:

(i) – x + y – z and – 2x

The multiplication of the given expression is calculated as,

(- x + y – z) × – 2x = – x × – 2x + y × – 2x – z × – 2x

On further calculation, we get

= 2x¹⁺¹ – 2xy + 2xz

= 2x² – 2xy + 2xz

Hence, the multiplication of (- x + y – z) and – 2x is 2x² – 2xy + 2xz

(ii) xy – yz and x²yz²

The multiplication of the given expression is calculated as,

(xy – yz) × (x²yz²) = xy × x²yz² – yz × x²yz²

We get,

= x¹⁺²y¹⁺¹z² – x²y¹⁺¹z¹⁺²

= x³y²z² – x²y²z³

Hence, the multiplication of (xy – yz) and x²yz² = x³y²z² – x²y²z³

(iii) 2xyz + 3xy and – 2y²z

The multiplication of the given expression is calculated as,

(2xyz + 3xy) × – 2y²z = 2xyz × – 2y²z + 3xy × – 2y²z

On further calculation, we get

= – 4xy¹⁺²z¹⁺¹ – 6xy¹⁺²z

= – 4xy³z² – 6xy³z

Hence, the multiplication of 2xyz + 3xy and – 2y²z = – 4xy³z² – 6xy³z

(iv) – 3xy² + 4x²y and – xy

The multiplication of the given expression is calculated as,

(- 3xy² + 4x²y) × – xy = 3x¹⁺¹y²⁺¹ – 4x²⁺¹y¹⁺¹

On calculation, we get

= 3x²y³ – 4x³y²

Hence, the multiplication of – 3xy² + 4x²y and – xy = 3x²y³ – 4x³y²

(v) 4xy and – x²y – 3x² y²

The multiplication of the given expression is calculated as,

(- x²y – 3x²y²) × 4xy = – x²y × 4xy – 3x²y² × 4xy

On further calculation, we get

= – 4x²⁺¹y¹⁺¹ – 12x²⁺¹y²⁺¹

= – 4x³y² – 12x³y³

Hence, the multiplication of 4xy and – x²y – 3x² y² = – 4x³y² – 12x³y³

6. Multiply:

(i) 3a + 4b – 5c and 3a

(ii) – 5xy and – xy² – 6x²y

Solution:

(i) 3a + 4b – 5c and 3a

The multiplication of the given expression is calculated as,

(3a + 4b – 5c) × 3a = 3a × 3a + 4b × 3a – 5c × 3a

On further calculation, we get

= 9a¹⁺¹ + 12ab – 15ac

= 9a² + 12ab – 15ac

Therefore, the multiplication of 3a + 4b – 5c and 3a = 9a² + 12ab – 15ac

(ii) – 5xy and – xy² – 6x²y

The multiplication of the given expression is calculated as,

– 5xy × (- xy² – 6x²y) = – 5xy × – xy² – 5xy × – 6x²y

On further calculation, we get

= 5x¹⁺¹y¹⁺² + 30x¹⁺²y¹⁺¹

= 5x²y³ + 30x³y²

Therefore, the multiplication of – 5xy and – xy² – 6x²y = 5x²y³ + 30x³y²

7. Multiply:

(i) x + 2 and x + 10

(ii) x + 5 and x – 3

(iii) x – 5 and x + 3

(iv) x – 5 and x – 3

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

Solution:

(i) x + 2 and x + 10

The given expression is calculated as follows

(x + 2) × (x + 10) = x × (x + 10) + 2 × (x + 10)

We get,

= x² + 10x + 2x + 20

= x² + 12x + 20

Hence, the multiplication of (x + 2) and (x + 10) = x² + 12x + 20

(ii) x + 5 and x – 3

The given expression is calculated as follows

(x + 5) × (x – 3) = x × (x – 3) + 5 × (x – 3)

On simplification, we get

= x² – 3x + 5x – 15

= x² + 2x – 15

Hence, the multiplication of (x + 5) and (x – 3) = x² + 2x – 15

(iii) x – 5 and x + 3

The given expression is calculated as follows

(x – 5) × (x + 3) = x × (x + 3) – 5 × (x + 3)

On further calculation, we get

= x² + 3x – 5x – 15

= x² – 2x – 15

Hence, the multiplication of (x – 5) and (x + 3) = x² – 2x – 15

(iv) x – 5 and x – 3

The given expression is calculated as,

(x – 5) × (x – 3) = x × (x – 3) – 5 × (x – 3)

On further calculation, we get

= x² – 3x – 5x + 15

= x² – 8x + 15

Hence, the multiplication of (x – 5) and (x – 3) = x² – 8x + 15

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

The given expression is calculated as,

(2x + y) × (x + 3y) = 2x × (x + 3y) + y × (x + 3y)

On simplification, we get

= 2x² + 6xy + xy + 3y²

= 2x² + 7xy + 3y²

Hence, the multiplication of (2x + y) and (x + 3y) = 2x² + 7xy + 3y²

8. Multiply:

(i) 3abc and – 5a²b²c

(ii) x – y + z and -2x

(iii) 2x – 3y – 5z and -2y

(iv) – 8xyz + 10 x²yz³ and xyz

(v) xyz and – 13xy²z + 15x²yz – 6xyz²

Solution:

(i) 3abc and – 5a²b²c

The given expression is calculated as follows,

3abc × – 5a²b²c = 3 × – 5 × a × a² × b × b² × c × c

On further calculation, we get

= – 15 × a¹⁺² × b¹⁺² × c¹⁺¹

= – 15 × a³ × b³ × c²

= – 15a³b³c²

Therefore, the multiplication of 3abc and – 5a²b²c = – 15a³b³c²

(ii) x – y + z and -2x

The given expression is calculated as follows,

(x – y + z) × – 2x = x × – 2x – y × – 2x + z × – 2x

On simplification, we get

= – 2x¹⁺¹ + 2xy – 2xz

= – 2x² + 2xy – 2xz

Therefore, the multiplication of x – y + z and -2x = – 2x² + 2xy – 2xz

(iii) 2x – 3y – 5z and -2y

The given expression is calculated as follows,

(2x – 3y – 5z) × – 2y = 2x × – 2y – 3y × – 2y – 5z × – 2y

On further calculation, we get

= – 4xy + 6y¹⁺¹ + 10yz

= – 4xy + 6y² + 10yz

Therefore, the multiplication of 2x – 3y – 5z and -2y = – 4xy + 6y² + 10yz

(iv) – 8xyz + 10 x²yz³ and xyz

The given expression is calculated as follows,

(- 8xyz + 10x²yz³) × xyz = – 8xyz × xyz + 10x²yz³ × xyz

On further calculation, we get

= – 8x¹⁺¹y¹⁺¹z¹⁺¹ + 10x²⁺¹y¹⁺¹z³⁺¹

= – 8x²y²z² + 10x³y²z⁴

Therefore, the multiplication of – 8xyz + 10 x²yz³ and xyz = – 8x²y²z² + 10x³y²z⁴

(v) xyz and – 13xy²z + 15x²yz – 6xyz²

The given expression is calculated as follows,

xyz × (- 13xy²z + 15x²yz – 6xyz²) = xyz × – 13xy²z + xyz ×15x²yz – xyz × 6xyz²

On simplification, we get

= – 13x¹⁺¹y¹⁺²z¹⁺¹ + 15x¹⁺²y¹⁺¹z¹⁺¹ – 6x¹⁺¹y¹⁺¹z¹⁺²

We get,

= – 13x²y³z² + 15x³y²z² – 6x²y²z³

Therefore, the multiplication of xyz and – 13xy²z + 15x²yz – 6xyz² = – 13x²y³z² + 15x³y²z² – 6x²y²z³

9. Find the product of:

(i) xy – ab and xy + ab

(ii) 2abc – 3xy and 2abc + 3xy

(iii) a + b – c and 2a – 3b

(iv) 5x – 6y – 7z and 2x + 3y

(v) 5x – 6y – 7z and 2x + 3y + z

Solution:

(i) xy – ab and xy + ab

The product of the given expression is calculated as,

(xy – ab) × (xy + ab) = xy × (xy + ab) – ab × (xy + ab)

On simplification, we get

= xy × xy + xy × ab – ab × xy – ab × ab

= x²y² + abxy – abxy – a²b²

= x²y² – a²b²

Hence, the product of (xy – ab) and (xy + ab) = x²y² – a²b²

(ii) 2abc – 3xy and 2abc + 3xy

The product of the given expression is calculated as,

(2abc – 3xy) × (2abc + 3xy)

= 2abc × (2abc + 3xy) – 3xy × (2abc + 3xy)

We get,

= 2abc × 2abc + 2abc × 3xy – 3xy × 2abc – 3xy × 3xy

= 4a²b²c² + 6abcxy – 6abcxy – 9x²y²

= 4a²b²c² – 9x²y²

Hence, the product of 2abc – 3xy and 2abc + 3xy = 4a²b²c² – 9x²y²

(iii) a + b – c and 2a – 3b

The product of the given expression is calculated as,

(a + b – c) × (2a – 3b)

= a × (2a – 3b) + b × (2a – 3b) – c × (2a – 3b)

= a × 2a – a × 3b + b × 2a – b × 3b – c × 2a + c × 3b

= 2a¹⁺¹ – 3ab + 2ab – 3b¹⁺¹ – 2ac + 3bc

We get,

= 2a² – ab – 3b² – 2ac + 3bc

Hence, the product of a + b – c and 2a – 3b = 2a² – ab – 3b² – 2ac + 3bc

(iv) 5x – 6y – 7z and 2x + 3y

The product of the given expression is calculated as,

(5x – 6y – 7z) × (2x + 3y)

= (5x – 6y – 7z) × 2x + (5x – 6y – 7z) × 3y

= 5x × 2x – 6y × 2x – 7z × 2x + 5x × 3y – 6y × 3y – 7z × 3y

We get,

= 10x² – 12xy – 14xz + 15xy – 18y² – 21yz

= 10x² + 3xy – 14xz – 18y² – 21yz

Hence, the product of 5x – 6y – 7z and 2x + 3y = 10x² + 3xy – 14xz – 18y² – 21yz

(v) 5x – 6y – 7z and 2x + 3y + z

The product of the given expression is calculated as,

(5x – 6y – 7z) × (2x + 3y + z)

= (5x – 6y – 7z) × 2x + (5x – 6y – 7z) × 3y + (5x – 6y – 7z) × z

= 5x × 2x – 6y × 2x – 7z × 2x + 5x × 3y – 6y × 3y – 7z × 3y + 5x × z – 6y × z – 7z × z

We get,

= 10x² – 12xy – 14xz + 15xy – 18y² – 21yz + 5xz – 6yz – 7z²

= 10x² – 12xy + 15xy – 14xz + 5xz – 18y² – 21yz – 6yz – 7z²

= 10x² + 3xy – 9xz – 18y² – 27yz – 7z²

Hence, the product of 5x – 6y – 7z and 2x + 3y + z = 10x² + 3xy – 9xz – 18y² – 27yz – 7z²

Exercise 19(D)

1. Divide:

(i) 3a by a

(ii) 15x by 3x

(iii) 16m by 4

(iv) 20x² by 5x

(v) 30p² by 10p²

Solution:

(i) 3a by a

3a ÷ a

This can be written as,

3a / a = (3 × a) / a

= 3

Hence, 3a ÷ a = 3

(ii) 15x by 3x

15x ÷ 3x

15x / 3x = (15 × x) / (3 × x)

This can be written as,

= (3 × 5 × x) / (3 × x)

We get,

= 5

Hence, 15x ÷ 3x = 5

(iii) 16m by 4

16m ÷ 4

16m / 4 = (16 × m) / 4

This can be written as,

= (4 × 4 × m) / 4

We get,

= 4m

Hence, 16m ÷ 4 = 4m

(iv) 20x² by 5x

20x² ÷ 5x

20x² / 5x = (20 × x²) / (5 × x)

This can be written as,

= (4 × 5 × x^2-1) / 5

= 4 × x

= 4x

Hence, 20x² ÷ 5x = 4x

(v) 30p² by 10p²

30p² ÷ 10p² = (30 × p²) / (10 × p²)

This can be written as,

= (3 × 10 × p^2-2) / 10

= 3 × p⁰

= 3 × 1

= 3

Hence, 30p² ÷ 10p² = 3

2. Simplify:

(i) 2x⁵ ÷ x²

(ii) 6a⁸ ÷ 3a³

(iii) 20xy ÷ – 5xy

(iv) – 24a²b²c² ÷ 6ab

(v) – 5x²y ÷ xy²

Solution:

(i) 2x⁵ ÷ x²

= (2 × x⁵) / x²

= 2 × x^5-2

= 2 × x³

We get,

= 2x³

Hence, 2x⁵ ÷ x² = 2x³

(ii) 6a⁸ ÷ 3a³

= (6 × a⁸) / (3 × a³)

This can be written as,

= (2 × 3 × a^8-3) / 3

We get,

= 2 × a⁵

= 2a⁵

Hence, 6a⁸ ÷ 3a³ = 2a⁵

(iii) 20xy ÷ – 5xy

= (20 × x × y) / (- 5 × x × y)

This can be written as,

= (4 × 5) / – 5

We get,

= – 4

Hence, 20xy ÷ – 5xy = – 4

(iv) – 24a²b²c² ÷ 6ab

= (- 24 × a² × b² × c²) / (6 × a × b)

This can be written as,

= (-4 × 6 × a^2-1 × b^2-1 × c²) / 6

We get,

= – 4 × a × b × c²

= – 4abc²

Hence, – 24a²b²c² ÷ 6ab = – 4abc²

(v) – 5x²y ÷ xy²

= (- 5 × x² × y) / (x × y²)

This can be written as,

= (- 5 × x^2-1) / y^2-1

We get,

= (- 5 × x) / y

= – 5x / y

Hence, – 5x²y ÷ xy² = – 5x / y

3. Divide:

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

(ii) – 15p⁶q⁸ by – 5p⁶q⁷

(iii) – 21m⁵n⁷ by 14m²n²

(iv) 36a⁴x⁵y⁶ by 4x²a³y²

(v) 20x³a⁶ by 5xy

Solution:

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

= – 3m / 4 ÷ 2m = – 3m / 4 × 1 / 2m

= – (3 × m) / (4 × 2 × m)

We get,

= – 3 / 8

Hence, (- 3m / 4) ÷ 2m = – 3 / 8

(ii) – 15p⁶q⁸ by – 5p⁶q⁷

– 15p⁶q⁸ ÷ – 5p⁶q⁷ = (- 15 × p⁶ × q⁸) / (- 5 × p⁶ × q⁷)

This can be written as,

= (3 × 5 × q^8-7) / 5

We get,

= 3 × q

= 3q

Hence, – 15p⁶q⁸ ÷ – 5p⁶q⁷ = 3q

(iii) – 21m⁵n⁷ by 14m²n²

– 21m⁵n⁷ ÷ 14m²n² = (- 21 × m⁵ × n⁷) / (14 × m² × n²)

This can be written as,

= (- 3 × 7 × m^5-2 × n^7-2) / (2 × 7)

= (- 3 × m³ × n⁵) / 2

We get,

= – 3m³n⁵ / 2

Hence, – 21m⁵n⁷ ÷ 14m²n² = – 3m³n⁵ / 2

(iv) 36a⁴x⁵y⁶ by 4x²a³y²

36a⁴x⁵y⁶ ÷ 4x²a³y² = (36 × a⁴ × x⁵ × y⁶) / (4 × x² × a³ × y²)

This can be written as,

= (4 × 9 × a^4-3 × x^5-2 × y^6-2) / 4

= 9 × a¹ × x³ × y⁴

We get,

= 9ax³y⁴

Hence, 36a⁴x⁵y⁶ ÷ 4x²a³y² = 9ax³y⁴

(v) 20x³a⁶ by 5xy

20x³a⁶ ÷ 5xy = (20 × x³ × a⁶) / (5 × x × y)

This can be written as,

= (4 × 5 × x^3-1 × a⁶) / (5 × y)

We get,

= (4 × x² × a⁶) / y

= 4x²a⁶ / y

Hence, 20x³a⁶ ÷ 5xy = 4x²a⁶ / y

4. Simplify:

(i) (- 15m⁵n²) / (- 3m⁵)

(ii) 35x⁴y² / – 15x²y²

(iii) (- 24x⁶y²) / (6x⁶y)

Solution:

(i) (-15m⁵n²) / (- 3m⁵) = (-15 × m⁵ × n²) / (- 3 × m⁵)

This can be written as,

= (3 × 5 × m^5-5 × n²) / 3

= 5 × m⁰ × n²

= 5 × 1 × n²

= 5n²

Hence, (-15m⁵n²) / (- 3m⁵) = 5n²

(ii) 35x⁴y² / – 15x²y²

35x⁴y² / – 15x²y² = (35 × x⁴ × y²) / (- 15 × x² × y²)

This can be written as,

= – (5 × 7 × x^4-2 × y^2-2) / (3 × 5)

= – (7 × x² × y⁰) / 3

We get,

= – 7x²y / 3

Hence, 35x⁴y² / – 15x²y² = – 7x²y / 3

(iii) (- 24x⁶y²) / (6x⁶y)

(- 24x⁶y²) / (6x⁶y) = (- 24 × x⁶ × y²) / (6 × x⁶ × y)

This can be written as,

= (- 4 × 6 × x^6-6 × y^2-1) / 6

= – 4 × x⁰ × y¹

= – 4y

Hence, (- 24x⁶y²) / (6x⁶y) = – 4y

5. Divide:

(i) 9x³ – 6x² by 3x

(ii) 6m² – 16m³ + 10m⁴ by – 2m

(iii) 15x³y² + 25x²y³ – 36x⁴y⁴ by 5x²y²

(iv) 36a³x⁵ – 24a⁴x⁴ + 18a⁵x³ by – 6a³x³

Solution:

(i) 9x³ – 6x² by 3x

9x³ – 6x² ÷ 3x = (9 × x³ – 6 × x²) / (3 × x)

Separating the terms, we get

= (9 × x³) / (3 × x) – (6 × x²) / (3 × x)

We get,

= 3 × x^3-1 – 2 × x^2-1

= 3x² – 2x

Hence, 9x³ – 6x² ÷ 3x = 3x² – 2x

(ii) 6m² – 16m³ + 10m⁴ by – 2m

6m² – 16m³ + 10m⁴ ÷ – 2m = (6 × m²– 16 × m³ + 10 × m⁴) / – 2 × m

Separating the terms, we get

= (6 × m² / – 2 × m) – (16 × m³) / (- 2 × m) + (10 × m⁴) / (- 2 × m)

= – 3 × m^2-1 + 8 × m^3-1 – 5 × m^4-1

= – 3 × m + 8 × m² – 5 × m³

We get,

= – 3m + 8m² – 5m³

Hence, 6m² – 16m³ + 10m⁴ ÷ – 2m = – 3m + 8m² – 5m³

(iii) 15x³y² + 25x²y³ – 36x⁴y⁴ by 5x²y²

15x³y² + 25x²y³ – 36x⁴y⁴ ÷ 5x²y² = (15x³y² + 25x²y³ – 36x⁴y⁴) / (5x²y²)

= (15 × x³ × y²) / (5 × x² × y²) + (25 × x² × y³) / (5 × x² × y²) – (36 × x⁴ × y⁴) / (5 × x² × y²)

On further calculation, we get

= 3 × x^3-2 × y^2-2 + 5 × x^2-2 × y^3-2 – (36 × x^4-2 × y^4-2) / 5

We get,

= 3 × x¹ × y⁰ + 5 × x⁰ × y¹ – (36 × x² × y²) / 5

= 3x + 5y – (36x²y²) / 5

Hence, 15x³y² + 25x²y³ – 36x⁴y⁴ ÷ 5x²y² = 3x + 5y – (36x²y²) / 5

(iv) 36a³x⁵ – 24a⁴x⁴ + 18a⁵x³ by – 6a³x³

36a³x⁵ – 24a⁴x⁴ + 18a⁵x³ ÷ (– 6a³x³) = (36a³x⁵ – 24a⁴x⁴ + 18a⁵x³) / – 6a³x³

= (36.a³.x⁵) / (-6.a³.x³) – (24.a⁴.x⁴) / (-6.a³.x³) + (18.a⁵.x³) / (-6.a³.x³)

We get,

= – 6.x^5-3 + 4.a^4-3.x^4-3 – 3.a^5-3

= – 6x² + 4ax – 3a²

Hence, 36a³x⁵ – 24a⁴x⁴ + 18a⁵x³ ÷ (– 6a³x³) = – 6x² + 4ax – 3a²