Sure, let's solve each part of the question step-by-step.
Given:
[tex]\[ x = 9 - 4\sqrt{5} \][/tex]
### Part (i) [tex]\(\frac{1}{x}\)[/tex]
To find [tex]\(\frac{1}{x}\)[/tex]:
[tex]\[ \frac{1}{x} = \frac{1}{9 - 4\sqrt{5}} \][/tex]
From the calculations, we find:
[tex]\[ \frac{1}{x} \approx 17.944271909999298 \][/tex]
### Part (ii) [tex]\( x + \frac{1}{x} \)[/tex]
Next, we need to find [tex]\( x + \frac{1}{x} \)[/tex]:
[tex]\[ x + \frac{1}{x} = (9 - 4\sqrt{5}) + 17.944271909999298 \][/tex]
Adding these values:
[tex]\[ x + \frac{1}{x} \approx 18.00000000000014 \][/tex]
### Part (iii) [tex]\( \sqrt{x} - \frac{1}{\sqrt{x}} \)[/tex]
To find [tex]\( \sqrt{x} - \frac{1}{\sqrt{x}} \)[/tex]:
First, find [tex]\( \sqrt{x} \)[/tex]:
[tex]\[ \sqrt{x} \approx \sqrt{9 - 4\sqrt{5}} \][/tex]
And then [tex]\( \frac{1}{\sqrt{x}} \)[/tex]:
[tex]\[ \frac{1}{\sqrt{x}} \approx \frac{1}{\sqrt{9 - 4\sqrt{5}}} \][/tex]
The difference between them:
[tex]\[ \sqrt{x} - \frac{1}{\sqrt{x}} \approx -4.000000000000017 \][/tex]
### Part (iv) [tex]\( x^2 + \frac{1}{x^2} \)[/tex]
To compute [tex]\( x^2 + \frac{1}{x^2} \)[/tex]:
First, find [tex]\( x^2 \)[/tex]:
[tex]\[ x^2 = (9 - 4\sqrt{5})^2 \][/tex]
Next, find [tex]\( \left( \frac{1}{x} \right)^2 \)[/tex]:
[tex]\[ \left( \frac{1}{x} \right)^2 \approx 17.944271909999298^2 \][/tex]
Summing these values:
[tex]\[ x^2 + \frac{1}{x^2} \approx 322.000000000005 \][/tex]
### Part (v) [tex]\( x^3 + \frac{1}{x^3} \)[/tex]
Finally, to find [tex]\( x^3 + \frac{1}{x^3} \)[/tex]:
Compute [tex]\( x^3 \)[/tex]:
[tex]\[ x^3 = (9 - 4\sqrt{5})^3 \][/tex]
And [tex]\( \left( \frac{1}{x} \right)^3 \)[/tex]:
[tex]\[ \left( \frac{1}{x} \right)^3 \approx 17.944271909999298^3 \][/tex]
Summing them:
[tex]\[ x^3 + \frac{1}{x^3} \approx 5778.000000000134 \][/tex]
So, the values are:
(i) [tex]\(\frac{1}{x} \approx 17.944271909999298\)[/tex]
(ii) [tex]\( x + \frac{1}{x} \approx 18.00000000000014\)[/tex]
(iii) [tex]\( \sqrt{x} - \frac{1}{\sqrt{x}} \approx -4.000000000000017\)[/tex]
(iv) [tex]\( x^2 + \frac{1}{x^2} \approx 322.000000000005\)[/tex]
(v) [tex]\( x^3 + \frac{1}{x^3} \approx 5778.000000000134\)[/tex]
