Question Number 180381 by Linton last updated on 11/Nov/22
$$\sqrt{\frac{\mathrm{8}^{\mathrm{12}} +\mathrm{4}^{\mathrm{13}} }{\mathrm{8}^{\mathrm{6}} +\mathrm{4}^{\mathrm{14}} }} \\ $$
Answered by floor(10²Eta[1]) last updated on 11/Nov/22
$$\sqrt{\frac{\mathrm{2}^{\mathrm{36}} +\mathrm{2}^{\mathrm{26}} }{\mathrm{2}^{\mathrm{18}} +\mathrm{2}^{\mathrm{28}} }}=\sqrt{\frac{\mathrm{2}^{\mathrm{26}} \left(\mathrm{2}^{\mathrm{10}} +\mathrm{1}\right)}{\mathrm{2}^{\mathrm{18}} \left(\mathrm{1}+\mathrm{2}^{\mathrm{10}} \right)}}=\sqrt{\frac{\mathrm{2}^{\mathrm{26}} }{\mathrm{2}^{\mathrm{18}} }}=\sqrt{\mathrm{2}^{\mathrm{8}} }=\mathrm{2}^{\mathrm{4}} =\mathrm{16} \\ $$
Commented by peter frank last updated on 11/Nov/22
$$\mathrm{thanks} \\ $$
Answered by a.lgnaoui last updated on 11/Nov/22
$$\sqrt{\frac{\left(\mathrm{4}×\mathrm{2}\right)^{\mathrm{12}} +\mathrm{4}^{\mathrm{13}} }{\left(\mathrm{2}×\mathrm{4}\right)^{\mathrm{6}} +\mathrm{4}^{\mathrm{14}} }}\:=\sqrt{\frac{\mathrm{4}^{\mathrm{12}} \left(\mathrm{2}^{\mathrm{12}+} +\mathrm{4}\right)}{\mathrm{4}^{\mathrm{6}} \left(\mathrm{2}^{\mathrm{6}} +\mathrm{4}^{\mathrm{8}} \right)}}=\mathrm{4}^{\mathrm{3}} \sqrt{\frac{\mathrm{2}^{\mathrm{2}} }{\mathrm{2}^{\mathrm{6}} }\left(\frac{\mathrm{2}^{\mathrm{10}} +\mathrm{1}}{\mathrm{2}^{\mathrm{10}} +\mathrm{1}}\right)}=\mathrm{4}^{\mathrm{2}} =\mathrm{16} \\ $$