Question

If x=7-4√3 find the value of √x+1/√x
A. O
B. 1
C. 4
D. -4​

Answer 1

Answer:

C

Step-by-step explanation:

Note that

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

Also

$7 \pm 4 \sqrt{3} \\ \\ = 3 + 4 \pm 2(2)( \sqrt{3} ) \\ \\ = ( \sqrt{3} ) {}^{2} + {2}^{2} \pm2(2) (\sqrt{3} ) \\ \\ = (2\pm \sqrt{3} ) {}^{2}$

So

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