If n is an even number, \[\frac{a}{c} > 0 \] must hold.

 

Examples:

 

\( \bullet \quad 2 \cdot \large { \sqrt[5]{ \frac{3}{16} } = \sqrt[5]{\frac{3 \cdot 2^5 }{16} }} = \sqrt[5]{6 } \)

\( \bullet \quad x \cdot y \cdot \large{\sqrt[3]{\frac{1}{x^2 \cdot y^2}} = \sqrt[3]{\frac{x \cdot y^3}{x^2 \cdot y^2}} }= \sqrt[3]{x \cdot y} \)

\( \bullet \quad \large-\frac{1}{3} \sqrt[4]{27 } = -\sqrt[4]{\frac{27}{3^4} } = -\sqrt[4]{\frac{27}{81} } = -\sqrt[4]{\frac{1}{3} } \)

 

Question 5

 

Given \[ A = (\sqrt{ 5}-3) \sqrt{7 + 3\sqrt{5 } } \], find A.

\[ \text{A)} -2\sqrt{ 2} \quad \text{B) } 2\sqrt{ 2} \quad \text{C) } -3 \quad \text{D) } 3 \quad \text{E)} 4 \]

 

Solution:

 

\[ \text{Since } \sqrt{5} – 3 < 0, \] \[ A = (\sqrt{5} – 3) \sqrt{7 + 3\sqrt{5}} \] \[ = – (3 – \sqrt{5}) \sqrt{7 + 3\sqrt{5}} \] \[ = – \sqrt{(3 – \sqrt{5})^2 (7 + 3\sqrt{5})} \] \[ = – \sqrt{(14 – 6\sqrt{5})(7 + 3\sqrt{5})} \] \[ = – \sqrt{2 (7 – 3\sqrt{5})(7 + 3\sqrt{5})} \] \[ = – \sqrt{2 \left[ 7^2 – (3\sqrt{5})^2 \right]} \] \[ = – \sqrt{2 \cdot 4} = -2\sqrt{2}. \]

\(\textbf{Answer: A} \)

 

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