Question : If $\left(y-\frac{1}{y}\right)=4$, find the value of $\left(y^6+\frac{1}{y^6}\right)$.
Option 1: 5774
Option 2: 4096
Option 3: 5776
Option 4: 5778
Correct Answer: 5778
Solution :
Given:
$\left(y-\frac{1}{y}\right)=4$
Cubing both sides, we get,
⇒ $\left(y-\frac{1}{y}\right)^{3}=4^{3}$
⇒ $y^{3}-\frac{1}{y^{3}}-3×y×\frac{1}{y}(y-\frac{1}{y})=64$
⇒ $y^{3}-\frac{1}{y^{3}}-3(4)=64$
⇒ $y^{3}-\frac{1}{y^{3}}=64+12=76$
Again, squaring both sides, we get
$(y^{3}-\frac{1}{y^{3}})^2=76^2=5776$
⇒ $\left(y^3)^2 + (\frac{1}{y^3}\right)^2-2×y^3×\frac{1}{y^3}=5776$
⇒ $\left(y^6+\frac{1}{y^6}\right)=5776+2=5778$
Hence, the correct answer is 5778.
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