Question

if x= 7+4 root 3 then find the value of root x +1/root x
pls answer fast

Answer 1

First of all, notice that 

7 + 4 sqrt(3) 

= 4 + 2 sqrt(3) + 2 sqrt(3) + 3 

= (2 + sqrt(3))^2 

Now, since 

√x + 1/√x = (x + 1) / sqrt(x), 

Substitute 7 + 4 sqrt(3) for x: 

= (7 + 4 sqrt(3) + 1) / sqrt(7 + 4 sqrt(3)) 

Collect terms: 

= ( 8 + 4 sqrt(3)) / (2 + sqrt(3)) 

Factor the numerator: 

= 4 [ ( 2 + sqrt(3) ) / (2 + sqrt(3)) ] 

= 4 (1) 

= 4

Answer 2

$x = 7 + 4 \sqrt{3} = 2^{2} + \sqrt{3} ^{2} + 2 (2) ( \sqrt{3}) = (2 + \sqrt{3})^{2}\\ \\ \frac{1}{x} = \frac{1}{7 + 4 \sqrt{3}} \\ \\ \frac{1}{x} = \frac{7 - 4 \sqrt{3}}{(7 + 4 \sqrt{3})(7 - 4 \sqrt{3})} \\ \\ \frac{1}{x} = \frac{7 - 4 \sqrt{3}}{49 - 48} \\ \\ \frac{1}{x} = 7 - 4 \sqrt{3} = 2^{2} + \sqrt{3} ^{2} - 2 (2) ( \sqrt{3}) = (2 - \sqrt{3})^{2}$

So,
$\sqrt{x} = 2 + \sqrt{3} \\ \\ \frac{1}{ \sqrt{x} } = 2 - \sqrt{3} \\ \\ \sqrt{x} + \frac{1}{ \sqrt{x} } = 4$

Answer
4