Absolutely, let's break down the problem step-by-step.
Given:
[tex]\[ x = 7 + 4\sqrt{3} \][/tex]
First, calculate the square root of [tex]\( x \)[/tex]:
[tex]\[ \sqrt{x} \approx 3.732050807568877 \][/tex]
Next, we find the reciprocal of the square root of [tex]\( x \)[/tex]:
[tex]\[ \frac{1}{\sqrt{x}} \approx 0.2679491924311227 \][/tex]
Now, we can determine the expression [tex]\(\sqrt{x} + \frac{1}{\sqrt{x}}\)[/tex]:
[tex]\[ \sqrt{x} + \frac{1}{\sqrt{x}} \approx 3.732050807568877 + 0.2679491924311227 = 4.0 \][/tex]
For part (ii), let's determine the square of [tex]\( x \)[/tex]:
[tex]\[ x^2 = (7 + 4\sqrt{3})^2 \approx 193.9948452238571 \][/tex]
Next, find the reciprocal of [tex]\( x^2 \)[/tex]:
[tex]\[ \frac{1}{x^2} \approx 0.0051547761428715625 \][/tex]
Now, sum these two values to get:
[tex]\[ x^2 + \frac{1}{x^2} \approx 193.9948452238571 + 0.0051547761428715625 = 193.99999999999997 \][/tex]
Thus, the final answers are:
(i) [tex]\(\sqrt{x} + \frac{1}{\sqrt{x}} = 4.0\)[/tex]
(ii) [tex]\(x^2 + \frac{1}{x^2} \approx 193.99999999999997\)[/tex]
