Question : If $\frac{x^8+1}{x^4}=14$, then the value of $\frac{x^{12}+1}{x^6}$ is:
Option 1: 16
Option 2: 14
Option 3: 52
Option 4: 64
Correct Answer: 52
Solution :
$\frac{x^8+1}{x^4}=14$
$⇒x^4+\frac{1}{x^4}=14$
$⇒x^4+\frac{1}{x^4}+2=14+2$
$⇒(x^2+\frac{1}{x^2})^2=(4)^2$
$⇒x^2+\frac{1}{x^2}=4$
Now, $\frac{x^{12}+1}{x^6}$
$=x^6+\frac{1}{x^6}$
$=(x^2)^3+\frac{1}{(x^2)^3}$
$=(x^2+\frac{1}{x^2})^3-3×x^2×\frac{1}{x^2}(x^2+\frac{1}{x^2})$
$=4^3-3(4)$
$=64-12=52$
Hence, the correct answer is 52.
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