RS Aggarwal solution class 8 chapter 6 Operations on Algebraic Expressions Exercise 6D

Exercise 6D

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Q1 | Ex-6D | Class 8 | RS AGGARWAL | chapter 6 | Operations on Algebraic Expressions

Question 1:

Find each of the following products:
(i) (x + 6)(x + 6)
(ii) (4x + 5y)(4x + 5y)
(iii) (7a + 9b)(7a + 9b)
(iv) 23x+45y23x+45y
(v) (x2 + 7)(x2 + 7)
(vi) 56a2+256a2+2

Answer 1:

(i) We have:

(x+6)(x+6)=(x+6)2=x2+62+2×x×6                [using (a+b)2=a2+b2+2ab]=x2+36+12x

(ii) We have:

(4x+5y)(4x+5y)=(4x+5y)2=4x2+5y2+2×4x×5y          [using (a+b)2=a2+b2+2ab]=16x2+25y2+40xy

(iii) We have:
(7a+9b)(7a+9b)=(7a+9b)2=7a2+9b2+2×7a×9b            [using (a+b)2=a2+b2+2ab]=49a2+81b2+126ab

(iv) We have:
23x+45y23x+45y=23x+45y2=23x2+45y2+2×23x×45y              [using (a+b)2=a2+b2+2ab]=49x2+1625y2+1615xy


(v) We have:
(x2+7)(x2+7)=(x2+7)2=x22+72+2×x2×7             [using (a+b)2=a2+b2+2ab]=x4+49+14x2

(vi) We have:
56a2+256a2+2=56a2+22=56a22+22+2×56a2×2               [using (a+b)2=a2+b2+2ab]=2536a4+4+103a2


Q2 | Ex-6D | Class 8 | RS AGGARWAL | chapter 6 | Operations on Algebraic Expressions

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Question 2:

Find each of the following products:
(i) (x − 4)(x − 4)
(ii) (2x − 3y)(2x − 3y)
(iii) 34x-56y34x-56y
(iv) x-3xx-3x
(v) 13x2-913x2-9
(vi) 12y2-13y12y2-13y

Answer 2:

(i) We have:
(x-4)(x-4)=(x-4)2=x2-2×x×4+42                   [using (a-b)2=a2-2ab+b2]=x2-8x+16

(ii) We have:
(2x-3y)(2x-3y)=(2x-3y)2=2x2-2×2x×3y+3y2                [using (a-b)2=a2-2ab+b2]=4x2-12xy+9y2

(iii) We have:
34x-56y34x-56y=34x-56y2=34x2-2×34x×56y+56y2           [using (a-b)2=a2-2ab+b2]=916x2-1512xy+2536y2

(iv) We have:
x-3xx-3x=x-3x2=x2-2×x×3x+3x2              [using (a-b)2=a2-2ab+b2]=x2-6+9x2

(v) We have:
13x2-913x2-9=13x2-92=13x22-2×13x2×9+92             [using (a-b)2=a2-2ab+b2]=19x4-6x2+81

(vi) We have:
12y2-13y12y2-13y=12y2-13y2=12y22-2×12y2×13y+13y2            [using (a-b)2=a2-2ab+b2]=14y4-13y3+19y2


Q3 | Ex-6D | Class 8 | RS AGGARWAL | chapter 6 | Operations on Algebraic Expressions

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Question 3:

Expand:
(i) (8a + 3b)2
(ii) (7x + 2y)2
(iii) (5x + 11)2
(iv) a2+2a2
(v) 3x4+2y92
(vi) (9x − 10)2
(vii) (x2yyz2)2
(viii) xy-yx2
(ix) 3m-45n2

Answer 3:

We shall use the identities (a+b)2 =a2 +b2 +2ab and (a-b)2 =a2 +b2 -2ab.

(i) We have:
(8a+3b)2=8a2+2×8a×3b+3b2=64a2+48ab+9b2

(ii)We have:
(7x+2y)2=7x2+2×7x×2y+2y2=49x2+28xy+4y2

(iii) We have :
(5x+11)2=5x2+2×5x×11+112=25x2+110x+121

(iv) We have:
a2+2a2=a22+2×a2×2a+2a2=a42+2+4a2

(v) We have:
3x4+2y92=3x42+2×3x4×2y9+2y92=9x162+13xy+4y281

(vi) We have:
(9x-10)29x2-2×9x×10+102=81x2-180x+100

(vii) We have:
(x2y-yz2)2x2y2-2×x2y×yz2+yz22=x4y2-2x2y2z2+y2z4

(viii) We have:
xy-yx2=xy2-2×xy×yx+yx2=x2y2-2+y2x2

(ix) We have:
3m-45n2=3m2-2×3m×45n+45n2=9m2-24mn5+1625n2


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Q4 | Ex-6D | Class 8 | RS AGGARWAL | chapter 6 | Operations on Algebraic Expressions

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Question 4:

Find each of the following products:
(i) (x + 3)(x − 3)
(ii) (2x + 5)(2x − 5)
(iii) (8 + x)(8 − x)
(iv) (7x + 11y)(7x − 11y)
(v) 5x2+34y25x2-34y2
(vi) 4x5-5y34x5+5y3
(vii) x+1xx-1x
(viii) 1x+1y1x-1y

Answer 4:

(i) We have:

(x+3)(x-3)=x2-9                                [using (a+b)(a-b)=a2-b2]

(ii) We have:

(2x+5)(2x-5)=4x2-25                              [using (a+b)(a-b)=a2-b2]

(iii) We have:

(8+x)(8-x)=64-x2                                 [using (a+b)(a-b)=a2-b2]

(iv) We have:

(7x+11y)(7x-11y)=49x2-121y2                        [using (a+b)(a-b)=a2-b2]

(v) We have:

5x2+34y25x2-34y2=25x4-916y4                       [using (a+b)(a-b)=a2-b2]

(vi) We have:

4x5-5y34x5+5y3=16x225-25y29                     [using (a+b)(a-b)=a2-b2)]

(vii) We have:
x+1xx-1x=x2-1x2                            [using (a+b)(a-b)=a2-b2]

(viii) We have:
1x+1y1x-1y=1x2-1y2                      [using (a+b)(a-b)=a2-b2]

(ix) We have:
2a+3b2a-3b=4a2-9b2                     [using (a+b)(a-b)=a2-b2]


Q5 | Ex-6D | Class 8 | RS AGGARWAL | chapter 6 | Operations on Algebraic Expressions

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Question 5:

Using the formula for squaring a binomial, evaluate the following:
(i) (54)2
(ii) (82)2
(iii) (103)2
(iv) (704)2

Answer 5:

We shall use the identity (a+b)2 =a2 +b2 +2ab.

(i)
542=(50+4)2=502+2×50×4+42=2500+400+16=2916

(ii)
822=(80+2)2=802+2×80×2+22=6400+320+4=6724

(iii)
1032=(100+3)2=1002+2×100×3+32=10000+600+9=10609

(iv)
7042=(700+4)2=7002+2×700×4+42=490000+5600+16=495616


Q6 | Ex-6D | Class 8 | RS AGGARWAL | chapter 6 | Operations on Algebraic Expressions

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Question 6:

Using the formula for squaring a binomial, evaluate the following:
(i) (69)2
(ii) (78)2
(iii) (197)2
(iv) (999)2

Answer 6:

We shall use the identity (a-b)2 = a2 +b2 -2ab.

(i)
692=(70-1)2=702-2×70×1+1=4900-140+1=4761

(ii)
782=(80-2)2=802-2×80×2+4=6400-320+4=6084

(iii)
1972=(200-3)2=2002-2×200×3+9=40000-1200+9=38809

(iv)
9992=(1000-1)2=10002-2×1000×1+1=1000000-2000+1=998001


Q7 | Ex-6D | Class 8 | RS AGGARWAL | chapter 6 | Operations on Algebraic Expressions

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Question 7:

Find the value of:
(i) (82)2 − (18)2
(ii) (128)2 − (72)2
(iii) 197 × 203
(iv) 198×198-102×10296
(v) (14.7 × 15.3)
(vi) (8.63)2 − (1.37)2

Answer 7:

We shall use the identity (a-b) (a+b)=a2 - b2.

(i)
(82)2-(18)2=(82-18)(82+18)=(64)(100)=6400

(ii)
(128)2-(72)2=(128-72)(128+72)=(56)(200)=11200

(iii)
197×203=(200-3)(200+3)=2002-32=40000-9=39991

(iv)
198×198-102×10296=1982-102296=(198-102)(198+102)96=(96)(300)96=300

(v)
(14.7×15.3)=(15-0.3)×(15+0.3)=(15)2-(0.3)2=225-0.09=224.91

(vi)
(8.63)2-(1.37)2=(8.63-1.37)(8.63+1.37)=(7.26)(10)=72.6


Q8 | Ex-6D | Class 8 | RS AGGARWAL | chapter 6 | Operations on Algebraic Expressions

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Question 8:

Find the value of the expression (9x2 + 24x + 16), when x = 12.

Answer 8:

9x2 + 24x + 16Given, x = 123x2 + 2 3x4 + 42  3x + 42312+4236 + 42402 = 1600

Therefore, the value of the expression (9x2 + 24x + 16), when x = 12, is 1600.


Q9 | Ex-6D | Class 8 | RS AGGARWAL | chapter 6 | Operations on Algebraic Expressions

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Question 9:

Find the value of the expression (64x2 + 81y2 + 144xy), when x = 11 and y=43.

Answer 9:

64x2+81y2+144xyGiven: x=11  y =438x2 + 9y2 + 28x9y8x +9y 28(11) +9(43)288 +122100210000
                                                                                                                                               
Therefore, the value of the expression (64x2 + 81y2 + 144xy), when x = 11 and y = 43, is 10000.y=43


Q10 | Ex-6D | Class 8 | RS AGGARWAL | chapter 6 | Operations on Algebraic Expressions

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Question 10:

Find the value of the expression (36x2 + 25y2 − 60xy), when x=23 and y=15.

Answer 10:

36x2+25y2-60xyx=23, y=15=6x2 + 5y2 - 26x5y=6x - 5y2=6(23) -5(15)2=4 - 12=329


Q11 | Ex-6D | Class 8 | RS AGGARWAL | chapter 6 | Operations on Algebraic Expressions

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Question 11:

If x+1x=4, find the values of
(i) x2+1x2 and
(ii) x4+1x4.

Answer 11:

(i)  x+1x= 4Squaring both the sides:x+1x2= 42x2+1x2+2x1x= 16x2+1x2+2 =16x2+1x2=16-2x2+1x2= 14

Therefore, the value of  x2+1x2 is 14.

x2 + 1x2 = 14Squaring both the sides:x4 + 1x4+2x21x2= 142x4 + 1x4+2 = 196x4 + 1x4 = 196-2x4 + 1x4=194

Therefore, the value of x4 + 1x4 is 194.


Q12 | Ex-6D | Class 8 | RS AGGARWAL | chapter 6 | Operations on Algebraic Expressions

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Question 12:

If x-1x=5, find the values of
(i) x2+1x2
(ii) x4+1x4.

Answer 12:

(i)  x-1x= 5Squaring both the sides:x-1x2= 52x2+1x2-2x1x= 25x2+1x2-2 =25x2+1x2=25+2x2+1x2= 27Therefore, the value of x2+1x2 is  27.

x2 + 1x2 = 27Squaring both the sides:x4 + 1x4+2x21x2= 272x4 + 1x4+2 = 729x4 + 1x4 = 729-2x4 + 1x4=727

Therefore, the value of x4 + 1x4 is 727.


Q13 | Ex-6D | Class 8 | RS AGGARWAL | chapter 6 | Operations on Algebraic Expressions

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Question 13:

Find the continued product:
(i) (x + 1)(x − 1)(x2 + 1)
(ii) (x − 3)(x + 3)(x2 + 9)
(iii) (3x − 2y)(3x + 2y)(9x2 + 4y2)
(iv) (2p + 3)(2p − 3)(4p2 + 9)

Answer 13:

i x+1x-1x2+1x2-x+x-1x2+1x2-1x2+1x22-122              according to the formula a2-b2 = a+ba-bx4-1.Therefore, the product of x+1x-1x2+1 is x4-1.

ii x-3x+3x2+9x2-32x2+9       according to the formula a2-b2 = a+ba-bx2-9x2+9x22-92                 according to the formula a2-b2 = a+ba-bx4-81Therefore, the product of x-3x+3x2+9 is x4-81.

iii 3x-2y3x+2y9x2+4y23x2-2y29x2+4y2        according to the formula a2-b2 = a+ba-b9x2-4y29x2+4y29x22-4y22                 according to the formula a2-b2 = a+ba-b81x4-16y4.Therefore, the product of 3x-2y3x+2y9x2+4y2 is 81x4-16y4.

iv 2p+32p-34p2+92p2-324p2+9       according to the formula a2-b2 = a+ba-b4p2-94p2+94p22-92                according to the formula a2-b2 = a+ba-b16p4-81.Therefore, the product of 2p+32p-34p2+9 is 16p4-81.


Q14 | Ex-6D | Class 8 | RS AGGARWAL | chapter 6 | Operations on Algebraic Expressions

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Question 14:

If x + y = 12 and xy = 14, find the value of (x2 + y2).

Answer 14:

x+y = 12On squaring both the sides:x+y2 = 122x2+y2+2xy = 144x2+y2 = 144 - 2xyGiven:  xy = 14x2+y2 = 144 - 214x2+y2 = 144 - 28x2+y2 = 116Therefore, the value of x2+y2 is 116.


Q15 | Ex-6D | Class 8 | RS AGGARWAL | chapter 6 | Operations on Algebraic Expressions

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Question 15:

If xy = 7 and xy = 9, find the value of (x2 + y2).

Answer 15:

x-y = 7On squaring both the sides:x-y2 = 72x2+y2-2xy = 49x2+y2 = 49 + 2xyGiven: xy = 9x2+y2 = 49 + 29x2+y2 = 49 + 18x2+y2 = 67.Therefore, the value of x2+y2 is 67.

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