Question : If $x^{3}+\frac{1}{x^{3}}=110$, then find the value of $x+\frac{1}{x}$:
Option 1: 2
Option 2: 3
Option 3: 4
Option 4: 5
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Correct Answer: 5
Solution :
Given: $x^{3}+\frac{1}{x^{3}}=110$
By using the algebraic identity,
$x^{3}+\frac{1}{x^{3}}=(x+\frac{1}{x})^3–3×x×\frac{1}{x}(x+\frac{1}{x})$
⇒ $110 = (x+\frac{1}{x})^3–3(x+\frac{1}{x})$
Now, putting the value of $(x+\frac{1}{x})=y$
⇒ $110 = y^3–3y$
By using trial and error, putting y = 5, we get
⇒ $110 = 5^3–3×5$
⇒ $110 = 125–15$
$110 =110$
Thus, $(x+\frac{1}{x})=5$
Hence, the correct answer is 5.
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