Question Number 221829 by fantastic last updated on 11/Jun/25
$$\sqrt{\mathrm{70}.\mathrm{71}.\mathrm{72}.\mathrm{73}+\mathrm{1}} \\ $$
Answered by aleks041103 last updated on 11/Jun/25
$$\mathrm{70}.\mathrm{71}.\mathrm{72}.\mathrm{73}+\mathrm{1}= \\ $$$$=\mathrm{70}.\mathrm{73}.\left(\mathrm{70}+\mathrm{1}\right)\left(\mathrm{73}−\mathrm{1}\right)+\mathrm{1}= \\ $$$$=\mathrm{70}.\mathrm{73}.\left(\mathrm{70}.\mathrm{73}+\mathrm{73}−\mathrm{70}−\mathrm{1}\right)+\mathrm{1}= \\ $$$$=\mathrm{70}.\mathrm{73}.\left(\mathrm{70}.\mathrm{73}+\mathrm{2}\right)+\mathrm{1}= \\ $$$$=\left(\mathrm{70}.\mathrm{73}\right)^{\mathrm{2}} +\mathrm{2}.\left(\mathrm{70}.\mathrm{73}\right).\mathrm{1}+\mathrm{1}^{\mathrm{2}} = \\ $$$$=\left(\mathrm{70}.\mathrm{73}+\mathrm{1}\right)^{\mathrm{2}} \\ $$$$\Rightarrow\sqrt{\mathrm{70}.\mathrm{71}.\mathrm{72}.\mathrm{73}+\mathrm{1}}=\mathrm{70}.\mathrm{73}+\mathrm{1}=\mathrm{5111} \\ $$
Commented by fantastic last updated on 11/Jun/25
$${thanks}\:{sir} \\ $$
Answered by Ghisom last updated on 11/Jun/25
$${x}\left({x}+\mathrm{1}\right)\left({x}+\mathrm{2}\right)\left({x}+\mathrm{3}\right)+\mathrm{1}=\left({x}^{\mathrm{2}} +\mathrm{3}{x}+\mathrm{1}\right)^{\mathrm{2}} \\ $$$${x}=\mathrm{70}\:\Rightarrow\:\mathrm{answer}\:\mathrm{is}\:\mathrm{5111} \\ $$
Commented by fantastic last updated on 11/Jun/25
$${thanks}\:{sir} \\ $$
Answered by Rasheed.Sindhi last updated on 11/Jun/25
$$\sqrt{\mathrm{70}.\mathrm{71}.\mathrm{72}.\mathrm{73}+\mathrm{1}} \\ $$$$=\sqrt{\frac{\mathrm{140}.\mathrm{142}.\mathrm{144}.\mathrm{146}}{\mathrm{2}^{\mathrm{4}} }+\mathrm{1}}\: \\ $$$$=\sqrt{\frac{\left(\mathrm{143}−\mathrm{3}\right)\left(\mathrm{143}−\mathrm{1}\right)\left(\mathrm{143}+\mathrm{1}\right)\left(\mathrm{143}+\mathrm{3}\right)+\mathrm{16}}{\mathrm{16}}}\: \\ $$$$=\frac{\mathrm{1}}{\mathrm{4}}\sqrt{\left(\mathrm{143}^{\mathrm{2}} −\mathrm{1}\right)\left(\mathrm{143}^{\mathrm{2}} −\mathrm{9}\right)+\mathrm{16}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{4}}\sqrt{\mathrm{143}^{\mathrm{4}} −\mathrm{10}.\mathrm{143}^{\mathrm{2}} +\mathrm{9}+\mathrm{16}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{4}}\sqrt{\mathrm{143}^{\mathrm{4}} −\mathrm{2}.\mathrm{5}.\mathrm{143}^{\mathrm{2}} +\mathrm{5}^{\mathrm{2}} } \\ $$$$=\frac{\mathrm{1}}{\mathrm{4}}\sqrt{\left(\mathrm{143}^{\mathrm{2}} −\mathrm{5}\right)^{\mathrm{2}} }\: \\ $$$$=\frac{\mathrm{1}}{\mathrm{4}}\left(\mathrm{143}^{\mathrm{2}} −\mathrm{5}\right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{4}}\left(\mathrm{20444}\right)=\mathrm{5111} \\ $$
Commented by fantastic last updated on 11/Jun/25
$${thanks}\:{sir} \\ $$
Answered by MathematicalUser2357 last updated on 25/Jun/25
$$\mathrm{5111} \\ $$