why does this: $${\sqrt{{\mathtt{7}}{\mathtt{\,-\,}}{\mathtt{4}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{3}}}}}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{7}}{\mathtt{\,\small\textbf+\,}}{\mathtt{4}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{3}}}}}}$$ equals 1?
$$\left({\mathtt{7}}{\mathtt{\,-\,}}{\mathtt{4}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{3}}}}\right){\mathtt{\,\times\,}}\left({\mathtt{7}}{\mathtt{\,\small\textbf+\,}}{\mathtt{4}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{3}}}}\right) = {\mathtt{49}}{\mathtt{\,-\,}}{\mathtt{16}}{\mathtt{\,\times\,}}{\mathtt{3}} = {\mathtt{1}}$$
$${\sqrt{{\mathtt{1}}}} = {\mathtt{1}}$$
radix !
$${\sqrt{{\mathtt{7}}{\mathtt{\,-\,}}{\mathtt{4}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{3}}}}}} = {\mathtt{0.267\: \!949\: \!192\: \!431\: \!122\: \!7}}$$
$${\sqrt{{\mathtt{7}}{\mathtt{\,\small\textbf+\,}}{\mathtt{4}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{3}}}}}} = {\mathtt{3.732\: \!050\: \!807\: \!568\: \!877\: \!3}}$$
$${\mathtt{0.267\: \!949\: \!192\: \!431\: \!122\: \!7}}{\mathtt{\,\times\,}}{\mathtt{3.732\: \!050\: \!807\: \!568\: \!877\: \!3}} = {\mathtt{1}}$$
$$\left({\mathtt{7}}{\mathtt{\,-\,}}{\mathtt{4}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{3}}}}\right){\mathtt{\,\times\,}}\left({\mathtt{7}}{\mathtt{\,\small\textbf+\,}}{\mathtt{4}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{3}}}}\right) = {\mathtt{49}}{\mathtt{\,-\,}}{\mathtt{16}}{\mathtt{\,\times\,}}{\mathtt{3}} = {\mathtt{1}}$$
$${\sqrt{{\mathtt{1}}}} = {\mathtt{1}}$$
radix !
Thanks Radix and anon,
I like this question, it is a difference of 2 squares question. (Just like Radix said )
$$\\\sqrt{7-4\sqrt3}\times \sqrt{7+4\sqrt3}\\\\
=\sqrt{(7-4\sqrt3)(7+4\sqrt3)}\\\\
=\sqrt{(7)^2-(4\sqrt3)^2}\\\\
=\sqrt{49-16\times 3}\\\\
=\sqrt{49-48}\\\\
=\sqrt{1}\\\\
=1$$