543+4153=we will solve it by optionsoption (1) 3+5(3+5)3=(3)3+(5)3+3(3)25+3(3)(5)2=33+55+95+153=183+145≠543+415option (2) 23+5(23+5)3=(23)3+(5)3+3(23)2(5)+3(23)(5)2=243+55+365+303=543+415i.e., 543+4153=23+5Hence option (2) is correct.

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$\sqrt[3]{54 \sqrt{3} + 41 \sqrt{5}} =$

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$\sqrt{3} + \sqrt{5}$

$2 \sqrt{3} + \sqrt{5}$

$\sqrt{3} + 2 \sqrt{5}$

$3 \sqrt{3} + \sqrt{5}$

answer is B.

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Detailed Solution

$\sqrt[3]{54 \sqrt{3} + 41 \sqrt{5}} =$
we will solve it by options
option (1) $\sqrt{3} + \sqrt{5}$
$(\sqrt{3} + \sqrt{5})^{3} = (\sqrt{3})^{3} + (\sqrt{5})^{3} + 3 (\sqrt{3})^{2} \sqrt{5} + 3 (\sqrt{3}) (\sqrt{5})^{2}$
$\begin{matrix} = 3 \sqrt{3} + 5 \sqrt{5} + 9 \sqrt{5} + 15 \sqrt{3} \\ = 18 \sqrt{3} + 14 \sqrt{5} \neq 54 \sqrt{3} + 41 \sqrt{5} \end{matrix}$
option (2) $2 \sqrt{3} + \sqrt{5}$
$\begin{matrix} (2 \sqrt{3} + \sqrt{5})^{3} = (2 \sqrt{3})^{3} + (\sqrt{5})^{3} + 3 (2 \sqrt{3})^{2} (\sqrt{5}) + 3 (2 \sqrt{3}) ({\sqrt{5})}^{2} \\ = 24 \sqrt{3} + 5 \sqrt{5} + 36 \sqrt{5} + 30 \sqrt{3} \\ = 54 \sqrt{3} + 41 \sqrt{5} \end{matrix}$
i.e., $\sqrt[3]{54 \sqrt{3} + 41 \sqrt{5}} = 2 \sqrt{3} + \sqrt{5}$
Hence option (2) is correct.