Question

examine whether the following number are rational 2 root 3 divided by 3 root under 52 - 4 root under 117​

Answer 1

$\frac{2 \sqrt{3} }{3 \sqrt{52} - 4 \sqrt{117} } \\ \\ \\ \frac{2 \sqrt{3}(3 \sqrt{52} + 4 \sqrt{117}) }{(3 \sqrt{52} - 4 \sqrt{117})(3 \sqrt{52} + 4 \sqrt{117})} \\ \\ \\ \frac{6 \sqrt{156} + 8 \sqrt{351} }{ {(3 \sqrt{52} )}^{2} - {(4 \sqrt{117} )}^{2} } \\ \\ \\ \frac{2(3 \sqrt{156} + 4 \sqrt{351} }{(9 \times 52) - (16 \times 117)} \\ \\ \\ \frac{2[(3 \sqrt{2 \times 2 \times 3 \times 13} ) + (4 \sqrt{3 \times 3 \times 3 \times 13} )]}{468 - 1872} \\ \\ \\ \frac{2[(6 \sqrt{39} ) + (12 \sqrt{39} )]}{ - 1404} \\ \\ \\ \frac{ - [(12 \sqrt{39}) + (24 \sqrt{39}) ]}{1404} \\ \\ \\ \frac{ - 1[12 \sqrt{39}(1 + 2)]}{1404} \\ \\ \\ \frac{ - 3(12 \sqrt{39} }{1404} = - 1( \frac{12 \sqrt{39} }{468} )$