Question : If $\left(x+\frac{1}{x}\right)=\sqrt{6}$ and $x1$, what is the value of $\left(x^8-\frac{1}{x^8}\right)$?
Option 1: $120\sqrt{3}$
Option 2: $128\sqrt{3}$
Option 3: $112\sqrt{3}$
Option 4: $108\sqrt{3}$

Question

Question : If $\left(x+\frac{1}{x}\right)=\sqrt{6}$ and $x>1$, what is the value of $\left(x^8-\frac{1}{x^8}\right)$?

Option 1: $120\sqrt{3}$

Option 2: $128\sqrt{3}$

Option 3: $112\sqrt{3}$

Option 4: $108\sqrt{3}$

Team Careers360 · Accepted Answer

Correct Answer: $112\sqrt{3}$

Solution : $\left(x+\frac{1}{x}\right)=\sqrt{6}$
Squaring both sides,
$\left(x^2+\frac{1}{x^2}+2\right)=6$
⇒ $x^4-4x^2+1=0$
⇒ $x^2=\frac{4\pm2\sqrt{3}}2$
⇒ $x^2 = 2+\sqrt{3}$ (Since $x>0$)
Squaring both sides,
⇒ $x^4 = 7+4\sqrt{3}$
Squaring both sides
⇒ $x^8 = 97+56\sqrt{3}$
So, $\frac{1}{x^8}=97-56\sqrt{3}$
So, $x^8-\frac{1}{x^8}=112\sqrt{3}$
Hence, the correct answer is $112\sqrt{3}$.

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Question : If (x+1x)=6 and x>1, what is the value of (x8−1x8)?Option 1: 1203Option 2: 1283Option 3: 1123Option 4: 1083

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Question : If $(x + \frac{1}{x}) = \sqrt{6}$ and $x > 1$ , what is the value of $(x^{8} - \frac{1}{x^{8}})$ ?
Option 1: $120 \sqrt{3}$
Option 2: $128 \sqrt{3}$
Option 3: $112 \sqrt{3}$
Option 4: $108 \sqrt{3}$