1.

If \(x + \frac{1}{x} = 3\), then the value of \({x^5} + \frac{1}{{{x^5}}}\) is1. 1332. 1433. 1234. None of the above

Answer» Correct Answer - Option 3 : 123

Formula used

(a + b)2 = a2 + b2 + 2ab

(a + b)3 = a3 + b3 + 3ab(a + b)

Calculation

\(x + \frac{1}{x} = 3\)

Squaring both sides of the equation

\((x + \frac{1}{x})^2 = 9\)

\(\Rightarrow x^2 + {1 \over x^2} + 2 = 9\)

\(\Rightarrow x^2 + {1 \over x^2} = 9 - 2 = 7\)

\(x + \frac{1}{x} = 3\)

Cubing both sides we get

\((x + \frac{1}{x})^3 = 27\)

\(\Rightarrow x^3 + {1 \over x^3} + 3( x + {1 \over x}) = 27\)

\(\Rightarrow x^3 + {1 \over x^3} = 27 - 9 = 18\)

Now, 

\((x^2 + {1 \over x^2 })(x^3 + {1 \over x^3}) = x^5 + {1 \over x^5} + x + {1 \over x}\)

\(\Rightarrow 7 \times 18 = x^5 + {1 \over x^5} + 3\)

\(\Rightarrow x^5 + {1 \over x^5} = 126 - 3 = 123\)


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