Question

if x=1/√(9+4√5),find x^3+7x^2+16x+7​

Answer 1

i hope this will help you.

Answer 2

5(√5) is answer

1/√(9+4√5)=x

4√5 can also be written as 2√20

.: 1/√(9+2√20)=x

After converting this into a+2√b form let's simplify this surd

.:1/√5+√4=x

On rationalization

1/√5+2×√5-2/√5-2 =x

.: √5-2/5-4=x. = √5-2=x

So

X=√5-2

x+2= √5

x^2+4x+4=5

x^2=1-4x

Sum:(x^3)+7(x^2)+16x+7

=x(x^2+7x)+16x+7

=x(1-4x+7x)+16x+7

=x(1+3x)+16x+7

=x+(3(x^2))+16x+7

=x+(3(1-4x))+16x+7

=x+3-12x+16x+7

=5x+10

=5(x+2)

=5(√5)

SORRY For skipping some steps...I don't want to make this solution lengthy.Hope you understand and find this helpful