Question

please send me solutions of all
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Answer 1

i) (103)²

$\boxed{ \tt \: Using \: the \: identity }$

$⟶ \red{ (a+b)^{2} = {a}^{2} + 2ab + {b}^{2} }$

$(103)^{2} = (100 + 3)^{2}$

$= (100)^{2} + 2 \times 100 \times 3 + (3)^{2}$

$= 10000 + 600 + 9$

$= 10609$

ii) 51²

$\boxed{ \tt \: Using \: the \: identity } \: \:$

$⟶ \red{ (a+b)^{2} = {a}^{2} + 2ab + {b}^{2} } \:$

$(51 )^{2} = (50 + 1)^{2}$

$= 50 ^{2} + 2 \times 50 \times 1+ (1)^{2}$

$= 2500 + 100 + 1 \:$

$= 2601$

iii) 98²

$\boxed{ \tt \: Using \: the \: identity } \:$

$⟶ \red{ (a - b)^{2} = {a}^{2} + 2ab - {b}^{2} } \:$

$(98)^{2} = (100 - 2)^{2}$

$= (100)^{2} - 2 \times 100 \times 2 + (2)^{2}$

$= 10000 - 400 + 4$

$= 9604$

iv) (9.9)²

$\boxed{ \tt \: Using \: the \: identity }$

$⟶ \red{ (a - b)^{2} = {a}^{2} + 2ab - {b}^{2} } \: \:$

$(9.9)^{2} = (10 - 0.1)^{2}$

$= (10)^{2} + (0.1)^{2} - 2 × 10 × 0.1$

$= 100 + 0.01 - 2$

$= 98 + 0.01$

$= 98.01$

v) 102×98

$\boxed{ \tt \: Using \: the \: identity } \:$

$⟶ \red{(a + b) (a - b) = {a}^{2} - {b}^{2} } \:$

$(102 \times 98)= (100+ 2) (100-2)$

$a = 100, b= 2$

$= (100+2)(100-2) = 100-24$

$= 10000-4$

$= 102 \times 98 = 9996$

vi) 9.9×10.1

$\boxed{ \tt \: Using \: the \: identity }$

$⟶ \red{(a + b) (a - b) = {a}^{2} - {b}^{2} } \:$

$(9.9 \times 10.1)= (10 - 0.1) (10 +0.1)$

$= (10)^{2} - (0.1)^{2}$

$= 100 - 0.01$

$= 99.99$

vii) 243²-157²

$\boxed{ \tt \: Using \: the \: identity } \:$

$⟶ \red{a^2-b^2=(a + b) (a - b)}$

$(243^2-157^2)=(243-157) (243+157)$

$= 86 \times 400$

$=34400$

viii) 139²-138²

$\boxed{ \tt \: Using \: the \: identity }$

$⟶ \red{a^2-b^2=(a + b) (a - b)} \:$

$(139^{2} -138^{2}) =(139+138)(139-138)$

$=277 \times 1$

$=277 \:$

Answer 2

★ Solution (1) :-

$\sf \longmapsto 103^2$

$\sf \longmapsto (a + b)^2 = a^2 + 2ab + b^2$

Here,

a = 100

b = 3

$\sf \longmapsto (100 + 3)^2 = 100^2 + 2(100)(3) + 3^2$

$\sf \longmapsto 100^2 + 2(100)(3) + 3^2$

$\sf \longmapsto 10000 + 600 + 9$

$\sf \longmapsto 10609$

★ Solution (2) :-

$\sf \longmapsto 51^2$

$\sf \longmapsto (a + b)^2 = a^2 + 2ab + b^2$

Here,

a = 50

b = 1

$\sf \longmapsto (50 + 1)^2 = 50^2 + 2(50)(1) + 1^2$

$\sf \longmapsto 50^2 + 2(50)(1) + 1^2$

$\sf \longmapsto 2500 + 100 + 1$

$\sf \longmapsto 2601$

★ Solution (3) :-

$\sf \longmapsto 98^2$

$\sf \longmapsto (a - b)^2 = a^2 - 2ab + b^2$

Here,

a = 100

b = 2

$\sf \longmapsto (100 - 2)^2 = 100^2 - 2(100)(2) + 2^2$

$\sf \longmapsto 100^2 - 2(100)(2) + 2^2$

$\sf \longmapsto 10000 - 400 + 4$

$\sf \longmapsto 9606$

★ Solution (4) :-

$\sf \longmapsto 9.9^2$

$\sf \longmapsto (a - b)^2 = a^2 - 2ab + b^2$

Here,

a = 10

b = 0.1

$\sf \longmapsto (10 - 0.1)^2 = 10^2 - 2(10)(0.1) + 0.1^2$

$\sf \longmapsto 10^2 - 2(10)(0.1) + 0.1^2$

$\sf \longmapsto 100 - 2 + 0.01$

$\sf \longmapsto 98.01$

★ Solution (5) :-

$\sf \longmapsto 102 \times 98$

$\sf \longmapsto (a + b)(a - b) = a^2 - b^2$

Here,

a = 100

b = 2

$\sf \longmapsto (100 + 2)(100 - 2) = 100^2 - 2^2$

$\sf \longmapsto 100^2 - 2^2$

$\sf \longmapsto 10000 - 4$

$\sf \longmapsto 9996$

★ Solution (6) :-

$\sf \longmapsto 9.9 \times 10.1$

$\sf \longmapsto (a - b)(a + b) = a^2 - b^2$

Here,

a = 10

b = 0.1

$\sf \longmapsto (10 - 0.1)(10 + 0.1) = 10^2 - 0.1^2$

$\sf \longmapsto 10^2 - 0.1^2$

$\sf \longmapsto 100 - 0.01$

$\sf \longmapsto 99.99$

★ Solution (7) :-

$\sf \longmapsto 243^2 - 157^2$

$\sf \longmapsto a^2 - b^2 = (a + b)(a - b)$

Here,

a = 243

b = 157

$\sf \longmapsto 243^2 - 157^2 = (243 + 157)(243 - 157)$

$\sf \longmapsto (243 + 157)(243 - 157)$

$\sf \longmapsto (400)(86)$

$\sf \longmapsto 34400$

★ Solution (8) :-

$\sf \longmapsto 139^2 - 138^2$

$\sf \longmapsto a^2 - b^2 = (a + b)(a - b)$

Here,

a = 139

b = 138

$\sf \longmapsto 139^2 - 138^2 = (139 + 138)(139 - 138)$

$\sf \longmapsto (239 + 138)(239 - 138)$

$\sf \longmapsto (377)(1)$

$\sf \longmapsto 377$

Hence solved !!