Question : If $x^{2} - \sqrt{7} x + 1 = 0$ , then what is the value of $x^{5} + \frac{1}{x^{5}} ?$
Option 1: $19 \sqrt{7}$
Option 2: $21 \sqrt{7}$
Option 3: $25 \sqrt{7}$
Option 4: $27 \sqrt{7}$

Question

Question : If $x^{2} - 7 x + 1 = 0$ and $0 < x < 1$ , what is the value of $x^{2} - \frac{1}{x^{2}} ?$

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Answer 1

Correct Answer: $19 \sqrt{7}$

Solution : Given: $x^{2} - \sqrt{7} x + 1 = 0$
Dividing by $x$ , we get:
⇒ $x + \frac{1}{x} = \sqrt{7}$
Now, Squaring both sides, we get:
⇒ $(x + \frac{1}{x})^{2} = (\sqrt{7})^{2}$
⇒ $x^{2} + \frac{1}{x^{2}} + 2 = 7$
⇒ $x^{2} + \frac{1}{x^{2}} = 5$
Similarly, Cubing both sides, we get:
⇒ $(x + \frac{1}{x})^{3} = (\sqrt{7})^{3}$
⇒ $x^{3} + \frac{1}{x^{3}} + 3 (x \times \frac{1}{x}) (x + \frac{1}{x}) = 7 \sqrt{7}$
⇒ $x^{3} + \frac{1}{x^{3}} + 3 \sqrt{7} = 7 \sqrt{7}$
⇒ $x^{3} + \frac{1}{x^{3}} = 4 \sqrt{7}$
Now, $x^{5} + \frac{1}{x^{5}} = (x^{3} + \frac{1}{x^{3}}) \times (x^{2} + \frac{1}{x^{2}}) - (x + \frac{1}{x})$
= $4 \sqrt{7} \times 5 - \sqrt{7}$
= $20 \sqrt{7} - \sqrt{7}$
= $19 \sqrt{7}$
Hence, the correct answer is $19 \sqrt{7}$ .

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Question : If x2−7x+1=0, then what is the value of x5+1x5?Option 1: 197Option 2: 217Option 3: 257Option 4: 277

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Question : If $x^{2} - \sqrt{7} x + 1 = 0$ , then what is the value of $x^{5} + \frac{1}{x^{5}} ?$
Option 1: $19 \sqrt{7}$
Option 2: $21 \sqrt{7}$
Option 3: $25 \sqrt{7}$
Option 4: $27 \sqrt{7}$