RD Sharma solution class 8 chapter 6 Algebraic Expressions and Identity Exercise 6.5

Exercise 6.5

Page-6.30



Question 1:

Multiply:
(5x + 3) by (7x + 2)

Answer 1:

To multiply, we will use distributive law as follows:

5x+37x+2=5x7x+2+37x+2=5x×7x+5x×2+3×7x+3×2=35x2+10x+21x+6=35x2+10x+21x+6=35x2+31x+6

Thus, the answer is 35x2+31x+6.


Question 2:

Multiply:
(2x + 8) by (x − 3)

Answer 2:

To multiply the expressions, we will use the distributive law in the following way:

2x+8x-3=2xx-3+8x-3=2x×x-2x×3+8x-8×3=2x2-6x+8x-24=2x2-6x+8x-24=2x2+2x-24

Thus, the answer is 2x2+2x-24.


Question 3:

Multiply:
(7x + y) by (x + 5y)

Answer 3:

To multiply, we will use distributive law as follows:

7x+yx+5y=7xx+5y+yx+5y=7x2+35xy+xy+5y2=7x2+36xy+5y2

Thus, the answer is 7x2+36xy+5y2.


Question 4:

Multiply:
(a − 1) by (0.1a2 + 3)

Answer 4:

To multiply, we will use distributive law as follows:

a-10.1a2+3=0.1a2a-1+3a-1=0.1a3-0.1a2+3a-3

Thus, the answer is 0.1a3-0.1a2+3a-3.


Question 5:

Multiply:
(3x2 + y2) by (2x2 + 3y2)

Answer 5:

To multiply, we will use distributive law as follows:

3x2+y22x2+3y2=3x22x2+3y2+y22x2+3y2=6x4+9x2y2+2x2y2+3y4=6x4+11x2y2+3y4

Thus, the answer is 6x4+11x2y2+3y4.


Question 6:

Multiply:
35x+12y by 56x+4y

Answer 6:

To multiply, we will use distributive law as follows:

35x+12y56x+4y=35x56x+4y+12y56x+4y=12x2+125xy+512xy+2y2=12x2+144+2560xy+2y2=12x2+16960xy+2y2

Thus, the answer is 12x2+16960xy+2y2.
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Question 7:

Multiply:
(x6y6) by (x2 + y2)

Answer 7:

To multiply, we will use distributive law as follows:

x6-y6x2+y2=x6x2+y2-y6x2+y2=x8+x6y2-y6x2+y8=x8+x6y2-y6x2-y8

Thus, the answer is x8+x6y2-y6x2-y8.


Question 8:

Multiply:
(x2 + y2) by (3a + 2b)

Answer 8:

To multiply, we will use distributive law as follows:

x2+y23a+2b=x23a+2b+y23a+2b=3ax2+2bx2+3ay2+2by2

Thus, the answer is 3ax2+2bx2+3ay2+2by2.


Question 9:

Multiply:
[−3d + (−7f)] by (5d + f)

Answer 9:

To multiply, we will use distributive law as follows:

-3d+-7f5d+f=-3d5d+f+-7f5d+f=-15d2-3df+-35df-7f2=-15d2-3df-35df-7f2=-15d2-38df-7f2

Thus, the answer is -15d2-38df-7f2.


Question 10:

Multiply:
(0.8a − 0.5b) by (1.5a − 3b)

Answer 10:

To multiply, we will use distributive law as follows:

0.8a-0.5b1.5a-3b=0.8a1.5a-3b-0.5b1.5a-3b=1.2a2-2.4ab-0.75ab+1.5b2=1.2a2-3.15ab+1.5b2

Thus, the answer is 1.2a2-3.15ab+1.5b2.


Question 11:

Multiply:
(2x2y2 − 5xy2) by (x2y2)

Answer 11:

To multiply, we will use distributive law as follows:

2x2y2-5xy2x2-y2=2x2y2x2-y2-5xy2x2-y2=2x4y2-2x2y4-5x3y2+5xy4

Thus, the answer is 2x4y2-2x2y4-5x3y2+5xy4.


Question 12:

Multiply:
x7+x22by25+9x4

Answer 12:

To multiply the expressions, we will use the distributive law in the following way:

x7+x2225+9x4=x725+9x4+x2225+9x4=2x35+9x228+x25+9x38=2x35+45+28140x2+ 9x38=2x35+73x2140 +9x28
Thus, the answer is  2x35+73x2140+9x38


Question 13:

Multiply:
-a7+a29byb2-b23

Answer 13:

To multiply, we will use distributive law as follows:

-a7+a29b2-b23=-a7b2-b23+a29b2-b23=-ab14+ab221+a2b18-a2b227=-ab14+ab221+a2b18-a2b227

Thus, the answer is -ab14+ab221+a2b18-a2b227.


Question 14:

Multiply:
(3x2y − 5xy2) by 15x2 + 13y2

Answer 14:

To multiply, we will use distributive law as follows:

3x2y-5xy215x2+13y2=15x23x2y-5xy2+13y23x2y-5xy2=35x4y-x3y2+x2y3-53xy4

Thus, the answer is 35x4y-x3y2+x2y3-53xy4.


Question 15:

Multiply:
(2x2 − 1) by (4x3 + 5x2)

Answer 15:

To multiply, we will use distributive law as follows:

2x2-14x3+5x2=2x24x3+5x2-14x3+5x2=8x5+10x4-4x3-5x2

Thus, the answer is 8x5+10x4-4x3-5x2.


Question 16:

(2xy + 3y2) (3y2 − 2)

Answer 16:

To multiply, we will use distributive law as follows:

2xy+3y23y2-2=2xy3y2-2+3y23y2-2=6xy3-4xy+9y4-6y2=9y4+6xy3-6y2-4xy

Thus, the answer is 9y4+6xy3-6y2-4xy.


Question 17:

Find the following product and verify the result for x = − 1, y = − 2:
(3x − 5y) (x + y)

Answer 17:

To multiply, we will use distributive law as follows:

3x-5yx+y=3xx+y-5yx+y=3x2+3xy-5xy-5y2=3x2-2xy-5y2

3x-5yx+y=3x2-2xy-5y2.
Now, we put x = -1 and y = -2 on both sides to verify the result.

LHS=3x-5yx+y=3-1-5-2-1+-2=-3+10-3=7-3=-21

RHS=3x2-2xy-5y2=3-12-2-1-2-5-22=3×1-4-5×4=3-4-20=-21

Because LHS is equal to RHS, the result is verified.

Thus, the answer is 3x2-2xy-5y2.


Question 18:

Find the following product and verify the result for x = − 1, y = − 2:
(x2y  1) (3  2x2y)

Answer 18:

To multiply, we will  use distributive law as follows:

x2y-13-2x2y=x2y3-2x2y-1×3-2x2y=3x2y-2x4y2-3+2x2y=5x2y-2x4y2-3

x2y-13-2x2y=5x2y-2x4y2-3

Now, we put x = -1 and y = -2 on both sides to verify the result.

LHS = x2y-13-2x2y=-12-2-13-2-12-2=1×-2-13-2×1×-2=-2-13+4=-3×7=-21

RHS=5x2y-2x4y2-3=5-12-2-2-14-22-3=5×1×-2-2×1×4-3=-10-8-3=-21

Because LHS is equal to RHS, the result is verified.

Thus, the answer is 5x2y-2x4y2-3.


Question 19:

Find the following product and verify the result for x = − 1, y = − 2:
13x-y2513x+y25

Answer 19:

To multiply, we will use distributive law as follows:

13x-y2513x+y25=13x13x+y25-y2513x+y25=19x2+xy215-xy215+y425=19x2+xy215-xy215-y425=19x2-y425

 13x-y2513x+y25=19x2-y425

Now, we will put x = -1 and y = -2 on both the sides to verify the result.

LHS = 13x-y2513x+y25=13-1--22513-1+-225=-13-45-13+45=-1715715=-119225

RHS=19x2-y425=19-12--2425=19×1-1625=19-1625=-119225

Because LHS is equal to RHS, the result is verified.

Thus, the answer is 19x2-y425.


Question 20:

Simplify:
x2(x + 2y) (x − 3y)

Answer 20:

To simplify, we will proceed as follows:

x2x+2yx-3y=x2x+2yx-3y=x3+2x2yx-3y=x3x-3y+2x2yx-3y=x4-3x3y+2x3y-6x2y2=x4-x3y-6x2y2

Thus, the answer is x4-x3y-6x2y2.


Question 21:

Simplify:
(x2 − 2y2) (x + 4y) x2y2

Answer 21:

To simplify, we will proceed as follows:

x2-2y2x+4yx2y2=x2x+4y-2y2x+4yx2y2=x3+4x2y-2xy2-8y3x2y2=x5y2+4x4y3-2x3y4-8x2y5

Thus, the answer is x5y2+4x4y3-2x3y4-8x2y5.


Question 22:

Simplify:
a2b2(a + 2b)(3a + b)

Answer 22:

To simplify, we will proceed as follows:

a2b2a+2b3a+b=a2b2a+2b3a+b=a3b2+2a2b33a+b=3aa3b2+2a2b3+ba3b2+2a2b3=3a4b2+6a3b3+a3b3+2a2b4=3a4b2+7a3b3+2a2b4

Thus, the answer is 3a4b2+7a3b3+2a2b4.


Question 23:

Simplify:
x2(x − y) y2(x + 2y)

Answer 23:

To simplify, we will proceed as follows:

x2x-yy2x+2y=x2x-yy2x+2y=x3-x2yxy2+2y3=x3xy2+2y3-x2yxy2+2y3=x4y2+2x3y3-x3y3+2x2y4=x4y2+2x3y3-x3y3-2x2y4=x4y2+x3y3-2x2y4

Thus, the answer is x4y2+x3y3-2x2y4.


Question 24:

Simplify:
(x3 − 2x2 + 5x − 7)(2x − 3)

Answer 24:

To simplify, we will proceed as follows:

x3-2x2+5x-72x-3=2xx3-2x2+5x-7-3x3-2x2+5x-7=2x4-4x3+10x2-14x-3x3+6x2-15x+21
=2x4-4x3-3x3+10x2+6x2-14x-15x+21     (Rearranging)
=2x4-7x3+16x2-29x+21                              (Combining like terms)

Thus, the answer is 2x4-7x3+16x2-29x+21.


Question 25:

Simplify:
(5x + 3)(x − 1)(3x − 2)

Answer 25:

To simplify, we will proceed as follows:

5x+3x-13x-2=5x+3x-13x-2
=5xx-1+3x-13x-2              (Distributive law)
=5x2-5x+3x-33x-2=5x2-2x-33x-2=3x5x2-2x-3-25x2-2x-3=15x3-6x2-9x-10x2-4x-6=15x3-6x2-9x-10x2+4x+6
=15x3-6x2-10x2-9x+4x+6              (Rearranging)
=15x3-16x2-5x+6                              (Combining like terms)

Thus, the answer is 15x3-16x2-5x+6.


Question 26:

Simplify:
(5 − x)(6 − 5x)( 2 − x)

Answer 26:

To simplify, we will proceed as follows:

5-x6-5x2-x=5-x6-5x2-x
=56-5x-x6-5x2-x                (Distributive law)
=30-25x-6x+5x22-x=30-31x+5x22-x=230-31x+5x2-x30-31x+5x2=60-62x+10x2-30x+31x2-5x3
=60-62x-30x+10x2+31x2-5x3              (Rearranging)
=60-92x+41x2-5x3                                (Combining like terms)

Thus, the answer is 60-92x+41x2-5x3.


Question 27:

Simplify:
(2x2 + 3x − 5)(3x2 − 5x + 4)

Answer 27:

To simplify, we will proceed as follows:

2x2+3x-53x2-5x+4
=2x23x2-5x+4+3x3x2-5x+4-53x2-5x+4           (Distributive law)
=6x4-10x3+8x2+9x3-15x2+12x-15x2+25x-20
=6x4-10x3+9x3+8x2-15x2-15x2+12x+25x-20              (Rearranging)
=6x4-x3-22x2+36x-20                                                     (Combining like terms)

Thus, the answer is 6x4-x3-22x2+36x-20.


Question 28:

Simplify:
(3x − 2)(2x − 3) + (5x − 3)(x + 1)

Answer 28:

To simplify, we will proceed as follows:

3x-22x-3+5x-3x+1=3x-22x-3+5x-3x+1
=3x2x-3-22x-3+5xx+1-3x+1           (Distributive law)
=6x2-9x-4x+6+5x2+5x-3x-3
=6x2+5x2-9x-4x+5x-3x-3+6                                  (Rearranging)
=11x2-11x+3                                                                 (Combining like terms)

Thus, the answer is 11x2-11x+3.


Question 29:

Simplify:
(5x − 3)(x + 2) − (2x + 5)(4x − 3)

Answer 29:

To simplify, we will proceed as follows:

5x-3x+2-2x+54x-3=5x-3x+2-2x+54x-3
=5xx+2-3x+2-2x4x-3+54x-3            (Distributive law)
=5x2+10x-3x-6-8x2+6x-20x+15
=5x2-8x2+10x-3x+6x-20x-6+15                              (Rearranging)
=5x2-8x2+10x-3x+6x-20x-6+15=-3x2-7x+9                              (Combining like terms)

Hence, the answer is -3x2-7x+9.


Question 30:

Simplify:
(3x + 2y)(4x + 3y) − (2xy)(7x − 3y)

Answer 30:

To simplify, we will proceed as follows:

3x+2y4x+3y-2x-y7x-3y=3x+2y4x+3y-2x-y7x-3y
=3x4x+3y+2y4x+3y-2x7x-3y-y7x-3y            (Distributive law)
=12x2+9xy+8xy+6y2-14x2-6xy-7xy+3y2=12x2+9xy+8xy+6y2-14x2+6xy+7xy-3y2
=12x2-14x2+9xy+8xy+6xy+7xy+6y2-3y2                              (Rearranging)
=-2x2+30xy+3y2                                                                       (Combining like terms)

Thus, the answer is -2x2+30xy+3y2.


Question 31:

Simplify:
(x2 − 3x + 2)(5x − 2) − (3x2 + 4x − 5)(2x − 1)

Answer 31:

To simplify, we will  proceed as follows:

x2-3x+25x-2-3x2+4x-52x-1=x2-3x+25x-2-3x2+4x-52x-1
=5xx2-3x+2-2x2-3x+2-2x3x2+4x-5-1×3x2+4x-5            (Distributive law)
=5x3-15x2+10x-2x2-6x+4-6x3+8x2-10x-3x2-4x+5=5x3-15x2+10x-2x2+6x-4-6x3+8x2-10x-3x2-4x+5=5x3-15x2+10x-2x2+6x-4-6x3-8x2+10x+3x2+4x-5
=5x3-6x3-15x2-2x2-8x2+3x2+10x+6x+10x+4x-5-4                                (Rearranging)
=-x3-22x2+30x-9                                                                                            (Combining like terms)

Thus, the answer is -x3-22x2+30x-9.
 


Question 32:

Simplify:
(x3 − 2x2 + 3x − 4) (x −1) − (2x − 3)(x2x + 1)

Answer 32:

To simplify,we will proceed as follows:

x3-2x2+3x-4x-1-2x-3x2-x+1=x3-2x2+3x-4x-1-2x-3x2-x+1
=xx3-2x2+3x-4-1x3-2x2+3x-4-2xx2-x+1-3x2-x+1            (Distributive law)
=xx3-2x2+3x-4-1x3-2x2+3x-4-2xx2-x+1-3x2-x+1=x4-2x3+3x2-4x-x3+2x2-3x+4-2x3-2x2+2x-3x2+3x-3=x4-2x3+3x2-4x-x3+2x2-3x+4-2x3+2x2-2x+3x2-3x+3
=x4-2x3-2x3-x3+3x2+2x2+2x2+3x2-4x-3x-2x-3x+4+3                             (Rearranging)
=x4-5x3+10x2-12x+7                                                                                            (Combining like terms)

Thus, the answer is x4-5x3+10x2-12x+7.

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