Question Number 1700 by 112358 last updated on 01/Sep/15 | ||
$${Show}\:{that}\:\left(\mathrm{7}!\right)^{\frac{\mathrm{1}}{\mathrm{7}}} <\left(\mathrm{8}!\right)^{\frac{\mathrm{1}}{\mathrm{8}}} . \\ $$ $${Also}\:{show}\:{that}\: \\ $$ $$\sqrt{\mathrm{100001}}−\sqrt{\mathrm{100000}}<\frac{\mathrm{1}}{\mathrm{2}\sqrt{\mathrm{100000}}}\:. \\ $$ $$ \\ $$ | ||
Commented by123456 last updated on 02/Sep/15 | ||
$$\mathrm{0}<\mathrm{7}<\mathrm{8}\Rightarrow\mathrm{0}<\frac{\mathrm{1}}{\mathrm{8}}<\frac{\mathrm{1}}{\mathrm{7}} \\ $$ $$\mathrm{0}<\mathrm{7}!<\mathrm{8}! \\ $$ $$\left(\mathrm{7}!\right)^{\frac{\mathrm{1}}{\mathrm{8}}} <\left(\mathrm{7}!\right)^{\frac{\mathrm{1}}{\mathrm{7}}} \\ $$ $$\left(\mathrm{8}!\right)^{\frac{\mathrm{1}}{\mathrm{8}}} <\left(\mathrm{8}!\right)^{\frac{\mathrm{1}}{\mathrm{7}}} \\ $$ $$\left(\mathrm{7}!\right)^{\frac{\mathrm{1}}{\mathrm{7}}} <\left(\mathrm{8}!\right)^{\frac{\mathrm{1}}{\mathrm{7}}} \\ $$ $$\left(\mathrm{7}!\right)^{\frac{\mathrm{1}}{\mathrm{8}}} <\left(\mathrm{8}!\right)^{\frac{\mathrm{1}}{\mathrm{8}}} \\ $$ | ||
Commented by123456 last updated on 02/Sep/15 | ||
$${f}\left({x}\right)=\left({x}!\right)^{\frac{\mathrm{1}}{{x}}} \\ $$ | ||
Answered by 123456 last updated on 01/Sep/15 | ||
$$\sqrt{\mathrm{100001}}−\sqrt{\mathrm{100000}}=\frac{\left(\sqrt{\mathrm{100001}}−\sqrt{\mathrm{100000}}\right)\left(\sqrt{\mathrm{100001}}+\sqrt{\mathrm{100000}}\right)}{\sqrt{\mathrm{100001}}+\sqrt{\mathrm{100000}}} \\ $$ $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\frac{\mathrm{100001}−\mathrm{100000}}{\sqrt{\mathrm{100001}}+\sqrt{\mathrm{100000}}} \\ $$ $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\frac{\mathrm{1}}{\sqrt{\mathrm{100001}}+\sqrt{\mathrm{100000}}} \\ $$ $$\mathrm{0}<\mathrm{100000}<\mathrm{100001} \\ $$ $$\mathrm{0}<\sqrt{\mathrm{100000}}<\sqrt{\mathrm{100001}} \\ $$ $$\mathrm{0}<\sqrt{\mathrm{100000}}+\sqrt{\mathrm{100000}}<\sqrt{\mathrm{100001}}+\sqrt{\mathrm{100000}} \\ $$ $$\mathrm{0}<\mathrm{2}\sqrt{\mathrm{100000}}<\sqrt{\mathrm{100001}}+\sqrt{\mathrm{100000}} \\ $$ $$\mathrm{0}<\frac{\mathrm{1}}{\sqrt{\mathrm{100001}}+\sqrt{\mathrm{100000}}}<\frac{\mathrm{1}}{\mathrm{2}\sqrt{\mathrm{100000}}} \\ $$ | ||