Question : If $a+\frac{1}{a}=8$, then what is the value of $\frac{1}{a^4}+a^4 ?$
Option 1: 4098
Option 2: 3846
Option 3: 3842
Option 4: 4094
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Correct Answer: 3842
Solution :
$(a + \frac{1}{a}) = 8$
Squaring both sides, we get,
⇒ $ (a^2 + \frac{1}{a^2}) + 2 = 64$
⇒ $(a^2 + \frac{1}{a^2}) = 62$
Squaring both sides again, we get,
⇒ $a^4 + \frac{1}{a^4} + 2 = 3844$
⇒ $a^4 + \frac{1}{a^4} = 3844 - 2$
⇒ $a^4 + \frac{1}{a^4} = 3842$
Hence, the correct answer is 3842.
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