Question

if x = 3+2 root2 check whether x +1byx is rational or urrational

Answer 1

It is rational.

For simplification when finding the value of 1/x rationalize by 3−2(2)1/23−2(2)1/2 , i.e. Multiply both numerator and denominator by this. This will make the denominator one and you will get 3−2(2)1/23−2(2)1/2 this as numerator.

x=3+2(2)1/2x=3+2(2)1/2

(1/x)=1/[3+2(2)1/2](1/x)=1/[3+2(2)1/2]

Now rationalize

(1/x)=1/[3+2(2)1/2]∗([3−2(2)1/2]/[3−2(2)1/2](1/x)=1/[3+2(2)1/2]∗([3−2(2)1/2]/[3−2(2)1/2]

(1/x)=[3−2(2)1/2]/[32−2∗(21/2)2](1/x)=[3−2(2)1/2]/[32−2∗(21/2)2]

(1/x)=[3−2(2)1/2]/[9−8]=3−2(2)1/2(1/x)=[3−2(2)1/2]/[9−8]=3−2(2)1/2

x+(1/x)=3+2(2)1/2+3−2(2)1/2=6x+(1/x)=3+2(2)1/2+3−2(2)1/2=6

As 6 is rational hence x+1/x is rational