Let's solve the given problem step-by-step.
We start with the equation:
[tex]\[ x - \frac{1}{x} = 8. \][/tex]
First, we square both sides of the equation to simplify it further:
[tex]\[ \left( x - \frac{1}{x} \right)^2 = 8^2. \][/tex]
Expanding the left-hand side using the algebraic identity [tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex], we get:
[tex]\[ x^2 - 2 \left( x \cdot \frac{1}{x} \right) + \frac{1}{x^2} = 64. \][/tex]
Simplifying the expression:
[tex]\[ x^2 - 2 + \frac{1}{x^2} = 64. \][/tex]
Next, we isolate [tex]\( x^2 + \frac{1}{x^2} \)[/tex] by adding 2 to both sides:
[tex]\[ x^2 + \frac{1}{x^2} = 64 + 2. \][/tex]
[tex]\[ x^2 + \frac{1}{x^2} = 66. \][/tex]
Now, we need to find the value of [tex]\( x^2 + \frac{1}{x^2} - 8 \)[/tex]. We substitute the value we just found:
[tex]\[ x^2 + \frac{1}{x^2} - 8 = 66 - 8. \][/tex]
[tex]\[ x^2 + \frac{1}{x^2} - 8 = 58. \][/tex]
Therefore, the correct answer is:
(ii) 58
