Question

plz answer questions 12,13

Answer 1

Ans.(13)

2+√3/2-√3 = a+b√3

2+√3× (2+√3) / (2-√3)(2+√3)

(2+√3)^2 / (2)^2- (√3)^2

[(a+b)^2= a^2+b^2+2ab]. & [(a+b)(a-b)=a^2-b^2]

(2^2+ (√3)^2+2×2×√3) / 4-3

(4+3+4√3) / 1

7+4√3 = a+b√3

on comparing
a= 7, b= 4

value of a^2+b^2
= (7)^2+(4)^2
=49+16
=65

Answer 2

Hi friend,

(12)
a = 2-√5/2+√5
a = 2-√5/2+√5×2-√5/2-√5
a = (2-√5)(2-√5)/(2+√5)(2-√5)
a = (2-√5)²/(2²-√5²)
a = [2²+√5²-2(2)(√5)]/(4-5)
a = (4+5-4√5)/-1
a = -(9-4√5)
a = -9+4√5

a² = (-9+4√5)² = (-9)²+(4√5)²+2(-9)(4√5)
= 81+80-72√5 = 161-72√5

b = 2+√5/2-√5
b = 2+√5/2-√5×2+√5/2+√5
b = (2+√5)(2+√5)/(2-√5)(2+√5)
b = (2+√5)²/(2²-√5²)
b = [2²+√5²+2(2)(√5)]/(4-5)
b = (4+5+4√5)/-1
b = -(9+4√5)
b = -9-4√5

b² = (-9-4√5)² = (-9)²+(4√5)²-2(-9)(4√5)
= 81+80+72√5 = 161+72√5

a²-b²
= 161-72√5-(161+72√5)
= 161-72√5-161-72√5
= -144√5

(13)
2+√3/2-√3
= (2+√3)/(2-√3)×(2+√3)/(2+√3)
= (2+√3)²/(2-√3)(2+√3)
= [2²+√3²+2(2)(√3)]/(2²-√3²)
= 4+3+4√3/4-3
= 7+4√3 = a+b√3

By comparing, we can say that
a = 7 and b = 4

a²+b² = 7²+4² = 49+16 = 65

Hope it helps......