Question

Evaluate-

{[√(√5+2)+√(√5-2)]/√(√5+1)}-√(3-2√2)

Answer 1

Given, P = {√(√5 + 2) + √(√5 -2)}/√(√5 + 1) - {√3 - 2√2}

Now, {√[√5 + 2] +√[√5 – 2]}/√[√5 + 1]

= { √[√5 + 2] + √[√5 – 2] } √[√5 – 1] / (√[√5 + 1] √[√5 – 1])

= { √[(√5 + 2)(√5 – 1)] + √[(√5 – 2)(√5 – 1)] } / √[(√5 + 1)(√5 – 1)]

= { √[5 – √5 + 2√5 – 2] + √[5 – √5 – 2√5 + 2] } / √[5 – 1]

= { √[3 + √5] + √[7 – 3√5] } / 2

= { √[(6 + 2√5)/2] + √[(14 – 6√5)/2] } / 2

= { √[(1 + 5 + 2√5)/2] + √[(9 + 5 – 6√5)/2] } / 2

= { √[(1 + √5)² /2] + √[(3 – √5)² /2] } / 2

= { (1 + √5)/√2 + (3 – √5)/√2 } / 2

= { (1 + √5) + (3 – √5) } / (2√2)

= 4 / (2√2)

= √2

Again, √[3 – 2√2] = √[2 + 1 – 2√2]

= √[(√2 – 1)² ]

= √2 – 1

Now, P = { √[√5 + 2] +√[√5 – 2] } / √[√5 + 1] – √[3 – 2√2]

=> P = √2 – (√2 – 1)

=> P = √2 – √2 + 1

=> P = 1

So, the value of P is 1