Question:
If \(x = \sqrt{3} + \sqrt{5}\), find the value of \(x^3 – 3x\).
यदि \(x = \sqrt{3} + \sqrt{5}\) है, तो \(x^3 – 3x\) का मान ज्ञात कीजिए।
Options:
- \(15\sqrt{3} + 11\sqrt{5}\)
- \(6\sqrt{3} + 8\sqrt{5}\)
- \(2\sqrt{3} + 5\sqrt{5}\)
- \(4\sqrt{3} + 6\sqrt{5}\)
- \(15\sqrt{3} + 11\sqrt{5}\)
- \(6\sqrt{3} + 8\sqrt{5}\)
- \(2\sqrt{3} + 5\sqrt{5}\)
- \(4\sqrt{3} + 6\sqrt{5}\)
Correct Answer: a)
Explanation:
\(x^2 = (\sqrt{3} + \sqrt{5})^2 = 3 + 5 + 2\sqrt{15} = 8 + 2\sqrt{15}\).
\(x^3 = x(x^2) = (\sqrt{3} + \sqrt{5})(8 + 2\sqrt{15}) = 8\sqrt{3} + 2\sqrt{45} + 8\sqrt{5} + 2\sqrt{75} = 18\sqrt{3} + 14\sqrt{5}\).
\(x^3 – 3x = (18\sqrt{3} + 14\sqrt{5}) – 3(\sqrt{3} + \sqrt{5}) = 15\sqrt{3} + 11\sqrt{5}\).
\(x^2 = 8 + 2\sqrt{15}\)
\(x^3 = (\sqrt{3} + \sqrt{5})(8 + 2\sqrt{15}) = 18\sqrt{3} + 14\sqrt{5}\)
\(x^3 – 3x = (18\sqrt{3} + 14\sqrt{5}) – (3\sqrt{3} + 3\sqrt{5}) = 15\sqrt{3} + 11\sqrt{5}\)। विकल्प (a) सही है।