Question : If $a^{4} + \frac{1}{a^{4}} = 194$ , then what is the value of $a^{3} + \frac{1}{a^{3}} ?$
Option 1: 44
Option 2: 52
Option 3: 48
Option 4: 50

Question

Question : If $t^{2} - 4 t + 1 = 0,$ then the value of $t^{3} + \frac{1}{t^{3}}$ is:

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Answer 1

Correct Answer: 52

Solution : Given: $a^{4} + \frac{1}{a^{4}} = 194$
Adding 2 both sides, we get:
⇒ $a^{4} + \frac{1}{a^{4}} + 2 = 194 + 2$
Now,
⇒ $(a^{2} + \frac{1}{a^{2}})^{2} = 14^{2}$
⇒ $(a^{2} + \frac{1}{a^{2}}) = 14$
Again, adding 2 both sides, we get:
⇒ $a^{2} + \frac{1}{a^{2}} + 2 = 14 + 2$
⇒ $(a + \frac{1}{a})^{2} = 4^{2}$
⇒ $a + \frac{1}{a} = 4$
Now, cubing both sides, we get:
$a^{3} + \frac{1}{a^{3}} + 3 \times a \times \frac{1}{a} (a + \frac{1}{a}) = 64$
⇒ $a^{3} + {\frac{1}{a}}^{3} + 3 \times 4 = 64$
⇒ $a^{3} + \frac{1}{a^{3}} = 52$
Hence, the correct answer is 52.

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Question : If a4+1a4=194, then what is the value of a3+1a3?Option 1: 44Option 2: 52Option 3: 48Option 4: 50

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Question : If $a^{4} + \frac{1}{a^{4}} = 194$ , then what is the value of $a^{3} + \frac{1}{a^{3}} ?$
Option 1: 44
Option 2: 52
Option 3: 48
Option 4: 50