Question

If x=3+2√2 find the value of √x-1/√x

Answer 1

x=3+2√2
x+¹/√x=?

x+¹/√x= (3+2√2)+1/(√3+2√2)
(√3+2√2 here means 3+2√2 whole root)
=(3+2√2)+(1)²/(√3+2√2)²
(squaring both numerator and dinominator)
=(3+2√2)+1/3+2√2
=(3+2√2)+1.(3-2√3)/(3+2√2)(3-2√2)
=(3+2√2)+(3-2√2)/3²-(2√2)²
=(3+2√2)+(3-2√2)/(9-8)
=3+2√2+3-2√2
=3+3
=6

Answer 2

GIVEN
x= 3 +2√2
x= 2+1 +2√2*1 

we do this to form it like an identity
x= (√2)²+(1)² +(2)(√2)(1)

now it is looking like the identity a²+b²+2ab=(a+b)²
 x= (√2)²+(1)² +(2)(√2)(1)


⇒x= (√2+1)²

√x = (√2+1)

now
1/√x = 1/ (√2+1)

we rationalize the denominator 
1/√x = 1/ (√2+1)* (√2-1)/(√2-1)
1/√x = 1(√2-1)/ (√2+1)(√2-1)

⇒1/√x = (√2-1) / (√2)² - (1)²
1/√x = (√2-1) / 2 - 1
1/√x = (√2-1) / 1
1/√x = (√2-1) 

so
√x - 1/√x = (√2+1) - (√2-1)
                = √2+1 - √2 + 1
               =2