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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