Question : If $k^{4} + \frac{1}{k^{4}} = 47$ , then what is the value of $k^{3} + \frac{1}{k^{3}}$ ?
Option 1: 4.5
Option 2: 54
Option 3: 18
Option 4: 9

Question

Question : If $k + \frac{1}{k} = 3$ , then what is the value of $k^{2} + \frac{1}{k^{2}}$ ?

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Answer 1

Correct Answer: 18

Solution : Given, $k^{4} + \frac{1}{k^{4}} = 47$
Adding 2 on both sides,
⇒ $k^{4} + \frac{1}{k^{4}} + 2 = 47 + 2$
⇒ $(k^{2})^{2} + \frac{1}{(k^{2})^{2}} + 2 \times k^{2} \times \frac{1}{k^{2}} = 49$
⇒ $(k^{2} + \frac{1}{k^{2}})^{2} = 49$
⇒ $k^{2} + \frac{1}{k^{2}} = 7$
Again adding 2 on both sides,
$k^{2} + \frac{1}{k^{2}} + 2 = 7 + 2$
⇒ $(k + \frac{1}{k})^{2} = 9$
⇒ $k + \frac{1}{k} = 3$
cubing both sides, we get,
⇒ $k^{3} + \frac{1}{k^{3}} + 3 \times k \times \frac{1}{k} (k + \frac{1}{k}) = 27$
⇒ $k^{3} + \frac{1}{k^{3}} + 3 \times 3 = 27$
$∴ k^{3} + \frac{1}{k^{3}} = 27 - 9 = 18$
Hence, the correct answer is 18.

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Question : If k4+1k4=47, then what is the value of k3+1k3?Option 1: 4.5Option 2: 54Option 3: 18Option 4: 9

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Question : If $k^{4} + \frac{1}{k^{4}} = 47$ , then what is the value of $k^{3} + \frac{1}{k^{3}}$ ?
Option 1: 4.5
Option 2: 54
Option 3: 18
Option 4: 9