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
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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:
$a^3+\frac{1}{a^3}+3×a×\frac{1}{a}(a+\frac{1}{a})=64$
⇒ $a^3+\frac{1}{a}^3+3\times 4 = 64$
⇒ $a^3+\frac{1}{a^3}=52$
Hence, the correct answer is 52.
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