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Number systems — MATHS STD 9 — Question

Rajasthan BoardEnglish MediumSTD 9MATHSNumber systems1 Mark
Question
If $\text{x}=\sqrt[3]{2+\sqrt3},$ then $\text{x}^3+\frac{1}{\text{x}^3}=$:
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Answer

  1. 4
    Solution:
    $\text{x}=\sqrt[3]{2+\sqrt3}+\big(2+\sqrt3\big)^{\frac{1}{3}}$
    $\text{x}^3=\big\{\big({2+\sqrt3}\big)^{\frac{1}{\not3}}\big\}^{\not3}=\big(2+\sqrt3\big)$
    $\Rightarrow\frac{1}{\text{x}^3}=\frac{1}{2+\sqrt3}=\frac{1}{2+\sqrt3}\times\frac{2-\sqrt3}{2-\sqrt3}\\ \ =\frac{2-\sqrt3}{4-3}=2-\sqrt3$
    Now, $\text{x}^3+\frac{1}{\text{x}^3}=2+\sqrt3+2-\sqrt3=4$
    Hence, correct option is (b).

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