Question : If $x^{4} + \frac{1}{x^{4}} = 34$ , what is the value of $x^{3} - \frac{1}{x^{3}}$ ?
Option 1: 0
Option 2: 6
Option 3: 8
Option 4: 14

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Question : If $x - \frac{1}{x} = 5, x \neq 0$ , then what is the value of $\frac{x^{6} + 3 x^{3} - 1}{x^{6} - 8 x^{3} - 1} ?$

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Answer 1

Correct Answer: 14

Solution : Given: $x^{4} + \frac{1}{x^{4}} = 34$
Adding 2 to both sides, we get,
$\Rightarrow x^{4} + \frac{1}{x^{4}} + 2 = 34 + 2$
$\Rightarrow (x^{2} + \frac{1}{x^{2}})^{2} = (6)^{2}$
$\Rightarrow x^{2} + \frac{1}{x^{2}} = 6$
Subtracting 2 from both sides, we get,
$\Rightarrow x^{2} + \frac{1}{x^{2}} - 2 = 6 - 2$
$\Rightarrow (x - \frac{1}{x})^{2} = (2)^{2}$
$\Rightarrow x - \frac{1}{x} = 2$
Now, $(x - \frac{1}{x})^{3} = x^{3} - \frac{1}{x^{3}} - 3 \times x \times \frac{1}{x} (x - \frac{1}{x})$
$\Rightarrow 2^{3} = x^{3} - \frac{1}{x^{3}} - 3 \times 2$
$∴ x^{3} - \frac{1}{x^{3}} = 14$
Hence, the correct answer is 14.

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Question : If x4+1x4=34, what is the value of x3−1x3?Option 1: 0Option 2: 6Option 3: 8Option 4: 14

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Question : If $x^{4} + \frac{1}{x^{4}} = 34$ , what is the value of $x^{3} - \frac{1}{x^{3}}$ ?
Option 1: 0
Option 2: 6
Option 3: 8
Option 4: 14