## RD Sharma Class 8 Solutions Chapter 1 Rational Numbers Ex 1.7

These Solutions are part of RD Sharma Class 8 Solutions. Here we have given RD Sharma Class 8 Solutions Chapter 1 Rational Numbers Ex 1.7

Other Exercises

Question 1.
Divide :

Solution:

Question 2.
Find the value and express as a rational number in standard form :

Solution:

Question 3.
The product of two rational numbers is15. If one of the numbers is -10, find the other.
Solution:

Question 4.
The product of two rational numbers is$$\frac { -9 }{ 8 }$$ if one of the numbers is $$\frac { -4 }{ 15 }$$, find other.
Solution:

Question 5.
By what number should we multiply $$\frac { -1 }{ 6 }$$ so that the product may be $$\frac { -23 }{ 9 }$$ ?
Solution:

Question 6.
By what number should we multiply $$\frac { -15 }{ 28 }$$ so that the product may be $$\frac { -5 }{ 7 }$$ ?
Solution:

Question 7.
By what number should we multiply $$\frac { -8 }{ 13 }$$ so that the product may be 24 ?
Solution:

Question 8.
Bv what number should $$\frac { -3 }{ 4 }$$ multiplied in order to produce $$\frac { 2 }{ 3 }$$ ?
Solution:

Question 9.
Find (x +y) + (x – y), if

Solution:

Question 10.
The cost of 7 $$\frac { 2 }{ 3 }$$ metres of rope is Rs 12 $$\frac { 3 }{ 4 }$$.Find its cost per metre.
Solution:

Question 11.
The cost of 2 $$\frac { 1 }{ 3 }$$ metres of cloth is Rs. 75 $$\frac { 1 }{ 4 }$$Find the cost of cloth per metre.
Solution:

Question 12.
By what number should $$\frac { -33 }{ 16 }$$ be divided to get $$\frac { -11 }{ 4 }$$ ?
Solution:

Question 13.
Divide the sum of $$\frac { -13 }{ 5 }$$ and $$\frac { 12 }{ 7 }$$ by the product of $$\frac { -31 }{ 7 }$$ and $$\frac { -1 }{ 2 }$$.
Solution:

Question 14.
Divide the sum of $$\frac { 65 }{ 12 }$$ and $$\frac { 12 }{ 7 }$$ bv their difference.
Solution:

Question 15.
If 24 trousers of equal size can be prepared in 54 metres of cloth, what length of cloth is required for each trouser ?
Solution:

Hope given RD Sharma Class 8 Solutions Chapter 1 Rational Numbers Ex 1.7 are helpful to complete your math homework.

If you have any doubts, please comment below. Learn Insta try to provide online math tutoring for you.

## RD Sharma Class 8 Solutions Chapter 1 Rational Numbers Ex 1.8

These Solutions are part of RD Sharma Class 8 Solutions. Here we have given RD Sharma Class 8 Solutions Chapter 1 Rational Numbers Ex 1.8

Other Exercises

Question 1.
Find a rational number between -3 and 1.
Solution:

Question 2.
Find any five rational number less than 1.
Solution:

Question 3.
Find four rational numbers between $$\frac { -2 }{ 9 }$$ and $$\frac { 5 }{ 9 }$$ .
Solution:

Question 4.
Find two rational numbers between $$\frac { 1 }{ 5 }$$ and $$\frac { 1 }{ 2 }$$ .
Solution:

Question 5.
Find ten rational numbers between $$\frac { 1 }{ 4 }$$ and $$\frac { 1 }{ 2 }$$ .
Solution:

Question 6.
Find ten rational numbers between $$\frac { -2 }{ 5 }$$ and $$\frac { 1 }{ 2 }$$ .
Solution:

Question 7.
Find ten rational numbers between $$\frac { 3 }{ 5 }$$ and $$\frac { 3 }{ 4 }$$ .
Solution:

Hope given RD Sharma Class 8 Solutions Chapter 1 Rational Numbers Ex 1.8 are helpful to complete your math homework.

If you have any doubts, please comment below. Learn Insta try to provide online math tutoring for you.

## RD Sharma Class 8 Solutions Chapter 2 Powers MCQS

These Solutions are part of RD Sharma Class 8 Solutions. Here we have given RD Sharma Class 8 Solutions Chapter 2 Powers MCQS

Other Exercises

Choose the correct alternative in each of the following :

Question 1.
Square of $$\left( \frac { -2 }{ 3 } \right)$$

Solution:

Question 2.
Cube of $$\frac { -1 }{ 2 }$$ is

Solution:

Question 3.
Which of the following is not equal to

Solution:

Question 4.
Which of the following in not reciprocal of

Solution:

Question 5.
Which of the following numbers is not equal to $$\frac { -8 }{ 27 }$$ ?

Solution:

Question 6.

Solution:

Question 7.

Solution:

Question 8.

Solution:

Question 9.

Solution:

Question 10.

Solution:

Question 11.

Solution:

Question 12.

Solution:

Question 13.

Solution:

Question 14.

Solution:

Question 15.
For any two non-zero rational numbers a and b,a4+b4 is equal to
(a) (a + b)1
(b) (a + b)0
(c) (a + b)4
(d) (a + b)8
Solution:
(c) {∵ a4 + b4 = (a + b)4}

Question 16.
For any two rational numbers a and b, a5 x b5 is equal to
(a) (a x b)0
(b) (a x b)10

(c) (a x b)5
(d) (a x b)25

Solution:
(c) {∵ a5 x b5 = (a x b)5}

Question 17.
For a non-zero rational number a, a7 + a12 is equal to
(a) a5
(b) a-19

(c) a-5
(d) a19

Solution:
(c) {a5 a12 = a7-12 =a-5}

Question 18.
For a non-zero rational number a, (a3)-2 is equal to
(a) a6
(b) a-6
(c) a-9
(d) a1

Solution:
(b) {(a3)-2 = a3 x (-2)= a6}

Hope given RD Sharma Class 8 Solutions Chapter 2 Powers MCQS are helpful to complete your math homework.

If you have any doubts, please comment below. Learn Insta try to provide online math tutoring for you.

## RD Sharma Class 8 Solutions Chapter 1 Rational Numbers Ex 1.5

These Solutions are part of RD Sharma Class 8 Solutions. Here we have given RD Sharma Class 8 Solutions Chapter 1 Rational Numbers Ex 1.5

Other Exercises

Question 1.
Multiply:

Solution:

Question 2.
Multiply:

Solution:

Question 3.
Simplify each of the following and express the result as a rational number in standard form :

Solution:

Question 4.
Simplify :

Solution:

Question 5.
Simplify :

Solution:

Hope given RD Sharma Class 8 Solutions Chapter 1 Rational Numbers Ex 1.5 are helpful to complete your math homework.

If you have any doubts, please comment below. Learn Insta try to provide online math tutoring for you.

## RD Sharma Class 8 Solutions Chapter 1 Rational Numbers Ex 1.4

These Solutions are part of RD Sharma Class 8 Solutions. Here we have given RD Sharma Class 8 Solutions Chapter 1 Rational Numbers Ex 1.4

Other Exercises

Question 1.
Simplify each of the following and write as a rational number of the form :

Solution:

Question 2.
Express each of the following as a rational number of the form $$\frac { p }{ q }$$:

Solution:

Question 3.
Simplify :

Solution:

Hope given RD Sharma Class 8 Solutions Chapter 1 Rational Numbers Ex 1.4 are helpful to complete your math homework.

If you have any doubts, please comment below. Learn Insta try to provide online math tutoring for you.

## RD Sharma Class 8 Solutions Chapter 1 Rational Numbers Ex 1.3

These Solutions are part of RD Sharma Class 8 Solutions. Here we have given RD Sharma Class 8 Solutions Chapter 1 Rational Numbers Ex 1.3

Other Exercises

Question 1.
Subtract the first rational number from the second in each of the following :

Solution:

Question 2.
Evaluate each of the following :

Solution:

Question 3.
The sum of two numbers is $$\frac { 5 }{ 9 }$$. If one of the numbers is $$\frac { 1 }{ 3 }$$, find the other.
Solution:

Question 4.
The sum of two numbers is $$\frac { -1 }{ 3 }$$. If one of the numbers is $$\frac { -12 }{ 3 }$$, find the other.
Solution:

Question 5.
The sum of two numbers is $$\frac { -4 }{ 3 }$$. If one of the number is -5, find the
Solution:

Question 6.
The sum of two rational numbers is -8. If one of the numbers is $$\frac { -15 }{ 7 }$$ find the other.
Solution:

Question 7.
What should be added to so as to $$\frac { -7 }{ 8 }$$ get $$\frac { 5 }{ 9 }$$ ?
Solution:

Question 8.
What number should be added to $$\frac { -5 }{ 11 }$$ so as to get $$\frac { 26 }{ 3 }$$ ?
Solution:

Question 9.
What number should be added to $$\frac { -5 }{ 7 }$$ to get $$\frac { -2 }{ 3 }$$ ?
Solution:

Question 10.
What number should be subtracted from $$\frac { -5 }{ 3 }$$ to get $$\frac { 5 }{ 6 }$$ ?
Solution:

Question 11.
What number should be subtracted from $$\frac { 3 }{ 7 }$$ to get $$\frac { 5 }{ 4 }$$ ?
Solution:

Question 12.
What should be added to $$\left( \frac { 2 }{ 3 } +\frac { 3 }{ 5 } \right)$$ to get $$\frac { -12 }{ 15 }$$ ?
Solution:

Question 13.
What should be added to $$\left( \frac { 1 }{ 2 } +\frac { 1 }{ 3 } +\frac { 1 }{ 5 } \right)$$ to get 3 ?
Solution:

Question 14.
What should be subtracted from $$\left( \frac { 3 }{ 4 } -\frac { 2 }{ 3 } \right)$$ to get $$\frac { -1 }{ 6 }$$ ?
Solution:

Question 15.
Fill in the blanks :

Solution:

Hope given RD Sharma Class 8 Solutions Chapter 1 Rational Numbers Ex 1.3 are helpful to complete your math homework.

If you have any doubts, please comment below. Learn Insta try to provide online math tutoring for you.

## RD Sharma Class 8 Solutions Chapter 1 Rational Numbers Ex 1.2

These Solutions are part of RD Sharma Class 8 Solutions. Here we have given RD Sharma Class 8 Solutions Chapter 1 Rational Numbers Ex 1.2

Other Exercises

Question 1.
Verify commutativity of addition of rational numbers for each of the following pairs of rational numbers :

Solution:

Question 2.
Verify associativity of addition of rational numbers i.e., (A: + y) + z = x + (y + z), when :

Solution:

Question 3.
Write the additive inverse of each of the following rational numbers :

Solution:

Question 4.
Write the negative (additive inverse) of each of the following :

Solution:

Question 5.
Using commutativity and associativity of addition of rational numbers, express ‘iach of the following as a rational number :

Solution:

Question 6.
Re-arrange suitably and find the sum in each of the following :

Solution:

Hope given RD Sharma Class 8 Solutions Chapter 1 Rational Numbers Ex 1.2 are helpful to complete your math homework.

If you have any doubts, please comment below. Learn Insta try to provide online math tutoring for you.

## RD Sharma Class 8 Solutions Chapter 1 Rational Numbers Ex 1.1

These Solutions are part of RD Sharma Class 8 Solutions. Here we have given RD Sharma Class 8 Solutions Chapter 1 Rational Numbers Ex 1.1

Other Exercises

Question 1.

Solution:

Question 2.

Solution:

Question 3.
Simplify:

Solution:

Question 4.
Add and Express the sum as mixed fraction:

Solution:

Hope given RD Sharma Class 8 Solutions Chapter 1 Rational Numbers Ex 1.1 are helpful to complete your math homework.

If you have any doubts, please comment below. Learn Insta try to provide online math tutoring for you.

## RD Sharma Class 8 Solutions Chapter 8 Division of Algebraic Expressions Ex 8.6

These Solutions are part of RD Sharma Class 8 Solutions. Here we have given RD Sharma Class 8 Solutions Chapter 8 Division of Algebraic Expressions Ex 8.6

Other Exercises

Divide :

Question 1.
x2 – 5x + 6 by x – 3
Solution:

Question 2.
ax2 – ay2 by ax + ay
Solution:

Question 3.
x– y4 by x– y2
Solution:

Question 4.
acx2 + (bc + ad)x + bd by (ax + b)
Solution:

Question 5.
(a2 + 2ab + b2)- (a2 + 2ac + c2) by 2a + b + c
Solution:

Question 6.
$$\frac { 1 }{ 4 }$$ x– $$\frac { 1 }{ 2 }$$ x- 12 by $$\frac { 1 }{ 2 }$$ x – 4
Solution:

Hope given RD Sharma Class 8 Solutions Chapter 8 Division of Algebraic Expressions Ex 8.6 are helpful to complete your math homework.

If you have any doubts, please comment below. Learn Insta try to provide online math tutoring for you.

## RD Sharma Class 8 Solutions Chapter 7 Factorizations Ex 7.5

These Solutions are part of RD Sharma Class 8 Solutions. Here we have given RD Sharma Class 8 Solutions Chapter 7 Factorizations Ex 7.5

Other Exercises

Factorize each of the following expressions :
Question 1.
16x2-25y2
Solution:
16x2 – 25y2 = (4x)2 – (5y)2    {∵ a2 – b2 = (a + b) (a – b)}
= (4x + 5y) (4x – 5y)

Question 2.
27x2 12y2
Solution:
27x2 – 12y2 = 3 (9x2 – 4y2)  {∵ a2 -b2 = (a + b) (a – b)}
= 3 [(3x)2 – (2y)2]
= 3 (3x + 2y) (3x – 2y)

Question 3.
144a– 289b2
Solution:
144a2 – 289b2 = (12a)2 – (17b)2    { ∵ a2b2 = (a + b) (a – b}
= (12a+ 17b) (12a- 17b)

Question 4.
12m2 – 27
Solution:
12m2 – 27 = 3 (4m2 – 9)
= 3 {(2m)2-(3)2}   {∵ a2b2 = (a + b) (a – b)}
= 3 (2m + 3) (2m – 3)

Question 5.
125x2 – 45y2
Solution:
125x2 – 45y2 = 5 (25x2 – 9y2)
= 5 {(5x-)2 – (3y)2}    {∵ a2 – b2 = (a + b) (ab}
= 5 (5x +
3y) (5x – 3y)

Question 6.
144a2 – 169b2
Solution:
144a2 – 169b2 = (12a)2 – (13b)2    {∵ a2 -b2 = (a + b) (a – b)}
= (12a + 13b) (12a-13b)

Question 7.
(2a – b)2 – 16c2
Solution:
(2a – b)2 – 16c2 = (2a – b)2 – (4c)2   {∵ a2 – b2 = (a + b) (a – b)}
= (2a – b + 4c) (2a – b – 4c)

Question 8.
(x + 2y)2 – 4 (2x -y)2
Solution:
(x + 2y)2 – 4 (2x – y)2
= (x + 2y)2 – {2 (2x –y)}2
=
(x + 2y)2 – (4x – 2y)2        {∵ a2– b2 = (a + b) (a – b)}
=
(a + 2y + 4x – 2y) (x + 2y – 4x + 2y)
= 5x (-3x + 4y)

Question 9.
3a5 – 48a3
Solution:
3a5 – 48a3 = 3a3 (a2– 16)
= 3a3 {(a)2 – (4)2}        {∵ a2 – b2 = (a + b) (a – b)}
= 3a3 (a + 4) (a – 4)

Question 10.
a4 – 16b4
Solution:
a4 – 16b4 = (a2)2 – (4b2)2
= (a2 + 4b2) (a2 – 4b2)
= (a2 + 4b2) {(a)2 – (2b)2 }   { ∵ a2 – b2 = (a + b) (a – b)}
= (a2 + 4b2) (a + 2b) (a – 2b)

Question 11.
x8 – 1
Solution:
x8 – 1 = (x4)2 – (1)2
= (x4 + 1) (x4 – 1)
= (x4+ 1) I (x2)2 – (1)2}             { a2 – b2 = (a + b) (a – b)}
= (x4 + 1) (x2 + 1) (x2 – 1)
= (x4 + 1) (x2 + 1) {(x)2 – (1)2}
= (x4+ 1)(x2 + 1)(x+ 1)(x- 1)
= (x-1)(x+ 1) (x2 + 1) (x4 + 1)

Question 12.
64 – (a + 1)2
Solution:
64 – (a + 1)2 = (8)2 – (a + 1)2    {∵ a2 – b2 = (a + b) (a – b)}
= (8 + a + 1) (8 – a – 1)
= (9 + a) (7 – a)

Question 13.
36l2 – (m + n)2
Solution:
36l2 – (m + n)2 = (6l)2 – (m + n)2        {∵  a2 – b2 = (a + b) (a – b)}
= (6l + m + n) (6l – m – n)

Question 14.
25x4y4 – 1
Solution:
25x4y4 – 1 = (5x4y4)2 – (1)2         { ∵  a2 – b2 = (a + b) (a – b)}
= (5x4y4  + 1) (5x2y2  – 1)

Question 15.

Solution:

Question 16.
x3 – 144x
Solution:
x3 – 144x = x (x2 – 144)
= x {(x)2 – (12)2}       { a2 – b2 = (a + b) (a – b)}
=  x (x + 12) (x – 12)

Question 17.
(x – 4y)2 – 625
Solution:
(x – 4y)2 – 625
= (x – 4y)2 – (25)2     {∵ a2 – b2 = (a + b) (a – b)}
= (x – 4y + 25) (x -4y – 25)

Question 18.
9 (a – b)2 – 100 (x -y)2
Solution:
9(a-b)2– 100(x-y)2
= {3(a-b)}2-{10(x-y)}2      { a2 – b2 = (a + b) (a – b)}
= (3a – 3b)2 – (10x – 10y)2
= (3a – 3b + 10x – 10y) (3a – 3b – 10x + 10y)

Question 19.
(3 + 2a)2 – 25a2
Solution:
(3 + 2a)2 – 25a2
= (3 + 2a)2 – (5a)2      ( a2 – b2 = (a + b) (a – b)}
= (3 + 2a + 5a) (3 + 2a – 5a)
= (3 + 7a) (3 – 3a)
= (3 + 7a) 3 (1 – a)
= 3(1-a) (3 +7a)

Question 20.
(x + y)2 – (a – b)2
Solution:

Question 21.

Solution:

Question 22.
75a3b2 – 108ab4
Solution:
75a3b2 – 108ab4
= 3ab2 (25a2 – 36b2)
= 3ab2 {(5a)2 – (6b)2}         { a2 – b2 = (a + b) (a – b)}
= 3ab2 (5a + 6b) (5a – 6b)

Question 23.
x5– 16x3
Solution:
x5 – 16x3 = x3 (x2 – 16)
= x3 {(x)2 – (4)2} { a2 – b2 = (a + b) (a – b)}
= x3 (x + 4) (x – 4)

Question 24.

Solution:

Question 25.
256x5 – 81x
Solution:
256x5– 81x = x(256x4– 81)
= x {(16x2)2 – (9)2}      {∵ a2 – b2 = {a + b) (a – b)}
= x (16x2 + 9) (16x2 – 9)
= x (16x2 + 9) {(4x)2 – (3)2}
= x (16x2 + 9) (4x + 3) (4x-3)

Question 26.
a4 – (2b + c)4
Solution:
a4 – (2b + c)4
= (a2)2 – [(2b + c)2]2    { a2 – b2 = (a + b) (a – b)}
= {a2 + (2b + c)2} {a2 – (2b + c)2}
= {a2 + (2b + c)2} {(a)2 – (2b + c)2}
= {a2 + (2b + c)2} (a + 2b + c) (a -2b- c)

Question 27.
(3x + 4y)4 – x4
Solution:
(3x + 4y)4 – x4 – [(3x + 4y)2]2 – (x2)2
= [(3x + 4y)2 + x2] [(3x + 4y)2 – x2]       {∵  a2 – b2 = (a + b) (a – b)
= [(3x + 4y)2 + x2] [(3x + 4y + x) (3x + 4y – x)]
=   [(3x + 4y)2 + x2] (4x + 4y) (2x + 4y)
= [(3x + 4y)2 + x2] 4 (x + y) 2 (x + 2y)
= 8 (x + y) (x + 2y) [(3x + 4y)2 + x2]

Question 28.
p2q2 – p4q4
Solution:
p2q2– p4q4 =p2q2 (1 -p2q2)
=p2q2 [(1)2 – (pq)2]   { a2 – b2 = (a + b) (a – b)
= p2q2 (1 +pq) (1 -pq)

Question 29.
3x3y – 243xy3
Solution:
3x3y – 243xy3
= 3xy (x2 – 81y2)
= 3xy [(x)2 – (9y)2]
= 3xy (x + 9y) (x – 9y)

Question 30.
a4b4 – 16c4
Solution:
a4b4 – 16c4 = (a2b2)2 – (4c2)2
= (a2b2 + 4c2) (a2b2 – 4c2)
= (a2b2 + 4c2) [(ab)2 – (2c)2]      { a2 – b2 = (a + b) (a – b)
= (a2b2 + 4c2) (ab + 2c) (ab – 2c)

Question 31.
x4-625
Solution:
x4 – 625 = (x2)2 – (25)2   { a2 – b2 – (a + b) (a – b)
= (x2 + 25) (x2 – 25)
= (x2 + 25) [(x)2 – (5)2]
= (x2 + 25) (x + 5) (x – 5)

Question 32.
x4-1
Solution:
x4 – 1 = (x2)2 – (1)2 = (x2 + 1) (x2 – 1)
= (x2 + 1) [(x)2 – (1)2]
= (x2 + 1) (x + 1) (x – 1)

Question 33.
49 (a – b)2 -25 (a + b)2
Solution:
49 (a – by -25 (a + b)2
= [7 (a – b)]2 [5 (a + b)]2
= (7a – 7b)2 – (5a + 5b)2  { a2 – b2 = (a + b) (a – b)
= (7a -7b + 5a + 5b) (7a – 7b -5a- 5b)
=(12a – 2b)(2a – 12b)
= 2 (6a – b) 2 (a – 6b)
= 4 (6 a- b) (a – 6b)

Question 34.
x – y – x2 + y
Solution:
x-y-x2 + y2 = (x-y)-(x2-y2) {∵ a2 – b2 = (a + b) (a – b)
= {x-y)-(x + y)(x-y)
= (x-y)(1 – x – y)

Question 35.
16 (2x – 1)2 – 25y2
Solution:
16 (2x – 1)2 – 25y2
= [4 (2x – 1)]2 – (5y)2
= (8x – 4)2 – (5y)2
= (8x – 4 + 5y) (8x -4-5y)
= (8x + 5y – 4) (8x – 5y – 4)

Question 36.
4 (xy + 1)2 – 9 (x – 1)2
Solution:
4 (xy + 1)2 – 9 (x – 1)2
=
[2 (xy + 1)]2 – [3 (x – 1)]2
= (2xy + 2)2 – (3x – 3){∵ a2 – b2 = (a + b) (a – b)
= (2xy + 2 + 3x – 3) (2xy + 2 – 3x + 3)
= (2xy + 3x – 1) (2xy – 3x + 5)

Question 37.
(2x + 1)2 – 9x4
Solution:
(2x + 1)2 – 9x4 = (2x + 1)2 – (3x2)2    { a2 – b2 = (a + b) (a – b)
= (2x + 1 + 3x2) (2x + 1 – 3x2)
= (3x2 + 2x + 1) (-3x + 2x + 1)

Question 38.
x4 – (2y- 3z)2
Solution:
x4 – (2y – 3z)2 = (x2)2 – (2y – 3z)2
= (x2 + 2y- 3z) (x2 – 2y + 3z)

Question 39.
a2-b2 +a-b
Solution:
a2 – b2 + a – b
= (a + b) {a – b) + 1 (a – b)
= (a – b) (a + b + 1)

Question 40.
16a4 – b4
Solution:
16a4 – b4
= (4a2)2 – (b2)2            {   a2 – b2 = (a + b) (a – b)
= (4a2 + b2) (4a2 – b2)
= (4a2 + b2) {(2a)2 – (b)2}
= (4a2 + b2) (2a + b) (2a – b)

Question 41.
a4 – 16 (b – c)4
Solution:
a4 – 16 (b- c)4 = (a2)2 – [4 (b – c)2]{   a2 – b2 = (a + b) (a – b)
= [a2 + 4 (b – c)2] [a2 – 4 (b – c)2]
= [a2 + 4 (b – c)2] [(a)2 – [2 (b – c)]2]
= [a2 + 4 (b – c)2] [(a)2 – (2b – 2c)2]
= [a2 + 4 (b – c)2] (a + 2b – 2c) (a – 2b + 2c)

Question 42.
2a5 – 32a
Solution:
2a5 – 32a = 2a (a4 – 16)
= 2a [(a2)2 – (4)2]  {∵  a2 – b2 = (a + b) (a – b)
= 2a (a2 + 4) (a2 – 4)]
= 2a (a2 + 4) [(a)2 – (2)2]
= 2a (a2 + 4) (a + 2) (a – 2)

Question 43.
a4b4 – 81c4
Solution:
a4b4 – 81c4 = (a2b2)2 – (9c2)2
= (a2b2 + 9c2) (a2b2 – 9c2
{∵ a2 – b2 = (a + b) (a – b)
= (a2b2 + 9c2) {(ab)2 – (3c)2}
= (a2b2 + 9c2) (ab + 3c) (ab – 3c)

Question 44.
xy9-yx9
Solution:
xy9yx9 = xy (y8 – x8)
= xy [(y4)2 – (x4)2{∵  a2 – b2 = (a + b) (a – b)}
= xy(y4 + x4)(y4-x4)
= xy (y4 + x4) {(y2)2 – (x2)2}
= xy (y4 + x4) (y2 + x2) (y2 – x2)
= xy (y4 + x4) (y2 + x2) (y + x) (y – x)

Question 45.
x3 -x
Solution:
x3-x = x(x2 1)
= x [(x)2 – (1)2] = x (x + 1) (x – 1)

Question 46.
18a2x2 – 32
Solution:
18a2x2 – 32
= 2 [9a2x2 – 16]
= 2 [(3ax)2 – (4)2]   { a2 – b2 = (a + b) (a – b)
= 2 (3ax + 4) (3ax – 4)

Hope given RD Sharma Class 8 Solutions Chapter 7 Factorizations Ex 7.5 are helpful to complete your math homework.

If you have any doubts, please comment below. Learn Insta try to provide online math tutoring for you.

## RD Sharma Class 8 Solutions Chapter 8 Division of Algebraic Expressions Ex 8.5

These Solutions are part of RD Sharma Class 8 Solutions. Here we have given RD Sharma Class 8 Solutions Chapter 8 Division of Algebraic Expressions Ex 8.5

Other Exercises

Question 1.
Divide the first polynomial by the second polynomial in each of the following. Also, write the quotient and remainder.
(i) 3x2 + 4x + 5, x – 2
(ii) 10x2 – 7x + 8, 5x – 3
(iii) 5y3– 6y2 + 6y-1,5y-1
(iv)x4-x3 + 5x,x-1
(v) y4 +y2,y2-2
Solution:
(i) 3x2 + 4x + 5, x – 2
= 3x (x – 2) + 10x + 5
= 3x (x – 2) + 10 (x – 2) + 25
∴ Quotient = 3x + 10
Remainder = 25

(iii) 5y3 – 6y2 + 6y – 1, 5y – 1
= y(5y – 1) – 5y2 + 6y- 1
= y2 (5y – 1) -y (5y – 1) + 5y – 1
= y2 (5y- 1) -y (5y- 1) + 1 (5y- 1)
∴ Quotient = y2 – y + 1 and Remainder = 0
(iv) x4 – x3 + 5x, x – 1
= x3(x – 1) + 5x
= x3 (x – 1) + 5 (x – 1) + 5
∴ Quotient = x3 + 5, Remainder = 5
(v) y4+y2,y2– 2
= y2(y– 2) + 3y2
= y2 (y2 – 2) + 3 (y2 – 2) + 6
∴ Quotient =y2 + 3 and Remainder = 6

Question 2.
Find, whether or not the first polynomial is a factor of the second :
(i) x + 1, 2x2 + 5x + 4
(ii) y- 2, 3y3 + 5y2 + 5y + 2
(iii) 4x2 – 5, 4.x4 + 7x2 + 15
(iv) 4-z, 3z2 – 13z + 4
(v) 2a-3,10a2 – 9a – 5
(vi) 4y+1 ,8y2-2y + 1
Solution:
(i) x + 1, 2x2 + 5x + 4
2x2 + 5x + 4 = 2x (x + 1) + 3x + 4
= 2x (x + 1) + 3 (x + 1) + 1
∵ Remainder = 1
∴ x + 1 is not a factor of 2x2 + 5x + 4
(ii) y – 2, 3y3 + 5y2 + 5y + 2
3y3 + 5y2 + 5y + 2 = 3y2(y – 2)+11y2 + 5y + 2
= 3y2(y – 2)+11y (y – 2) + 27y + 2
= 3y2 (y – 2) + 11y (y – 2) + 27 (y – 2) + 56
∵ Remainder = 56
∴ y – 2 is not a factor of 3y3 + 5y2 + 5y + 2
(iii) 4x2 – 5, 4x4 + 7x2 + 15
4x4 + 7x2 + 15 = x2 (4x2 – 5) + 12x2 + 15
= x2 (4x2 – 5) + 3 (4x2 – 5) + 30
∵ Remainder = 30
∴ 4x2 – 5 is not a factor of 4x4 + 7x2 + 15
(iv) 4 – z, 3z2 – 13z + 4
3z2 – 13z + 4 = -3z (-z + 4) – z + 4
= -3z (-z + 4) + 1 (-z + 4)
∵ Remainder = 0
∴ 4 – z or – z + 4 is a factor of 3z2 – 13z + 4
(v) 2a – 3, 10a2 – 9a – 5
10a2 – 9a – 5 = 5a (2a – 3) + 6a – 5
= 5a (2a – 3) + 3 (2a – 3) + 4
∵ Remainder = 4
∴ 2a – 3 is not a factor of 10a2 – 9a – 5
(vi) 4y + 1, 8y2 – 2y + 1
8y2 – 2y + 1 = 2y (4y + 1) – 4y + 1
= 2y (4y + 1) – 1 (4y + 1) + 2
∵ Remainder = 2
∴ 4y + 1 is not a factor of 8y2 – 2y + 1

Hope given RD Sharma Class 8 Solutions Chapter 8 Division of Algebraic Expressions Ex 8.5 are helpful to complete your math homework.

If you have any doubts, please comment below. Learn Insta try to provide online math tutoring for you.