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27. If [tex]$x=9-4 \sqrt{5}$[/tex], find the values of:

(i) [tex]\frac{1}{x}[/tex]

(ii) [tex]x+\frac{1}{x}[/tex]

(iii) [tex]\sqrt{x}-\frac{1}{\sqrt{x}}[/tex]

(iv) [tex]x^2+\frac{1}{x^2}[/tex]

(v) [tex]x^3+\frac{1}{x^3}[/tex]

Sagot :

GinnyAnsweridn
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]
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