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If \( \Large x=\sqrt{3}+\sqrt{2} \), then the value of \( \Large \left(x^{3}+\frac{1}{x^{3}}\right) \) is

A) \( \Large 6\sqrt{3} \)
B) \( \Large 12\sqrt{3} \)
C) \( \Large 18\sqrt{3} \)
D) \( \Large 24\sqrt{3} \)

Correct Answer:



C) \( \Large 18\sqrt{3} \)

Description for Correct answer:
\( \Large x=\sqrt{3}+\sqrt{2} \)

Therefore, \( \Large \frac{1}{x}=\sqrt{3}-\sqrt{2} \)

\( \Large x^{3}+\frac{1}{x^{3}} \)

Therefore, \( \Large x^{3}= \left(\sqrt{3}+\sqrt{2}\right)^{3} \)

= \( \Large \left(\sqrt{3}\right)^{3}+\left(\sqrt{2}\right)^{3}+3 \times \sqrt{3} \times \sqrt{2} \left(\sqrt{3}+\sqrt{2}\right) \)

= \( \Large 3\sqrt{3}+2\sqrt{2}+3\sqrt{6} \left(\sqrt{3}+\sqrt{2}\right) \)

= \( \Large 3\sqrt{3}+2\sqrt{2}+9\sqrt{2}+6\sqrt{3} \)

\( \Large x^{3}=9\sqrt{3}+11\sqrt{2} \)

\( \Large \frac{1}{x^{3}}=9\sqrt{3}-11\sqrt{2} \)

\( \Large x^{3}+\frac{1}{x^{3}}=9\sqrt{3}+11\sqrt{2}+9\sqrt{3}-11\sqrt{2} \)

= \( \Large 18\sqrt{3} \)

Part of solved Elementary Mathematics questions and answers : >> Elementary Mathematics

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