1.

If \({x^2} - 3\sqrt 2 x + 1 = 0\), then the value of \({x^3} + \frac{1}{{{x^3}}}\) is:1. \(45\sqrt 2 \)2. \(54\sqrt 2 \)3. \(24\sqrt 6 \)4. \(36\sqrt 6\)

Answer» Correct Answer - Option 1 : \(45\sqrt 2 \)

Given:

\({x^2} - 3√ 2 x + 1 = 0\)

Formula used:

(a + b)3 = a3 + b3 + 3ab(a + b)

Calculation:

\({x^2} - 3√ 2 x + 1 = 0\)

⇒ x(x – 3√2 + 1/x) = 0

⇒ (x – 3√2 + 1/x) = 0

⇒ x + 1/x = 3√2      ----(1)

Cubing on both side

⇒ (x + 1/x)3 = (3√2)3

(a + b)3 = a3 + b3 + 3ab(a + b)

⇒ x3 + 1/x3 + 3(x + 1/x) = 54√2

⇒ x3 + 1/x3 + 3 × 3√2  = 54√2  (from equation 1)

⇒ x3 + 1/x3 = 54√2 – 9√2

⇒ x3 + 1/x3 = 45√2

∴ The value of x3 + 1/x3 is 45√2


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