Question

prove (without using a calculator ) $55!\ \textless \ 28^{55}$

Answer 1

$55! \ \textless \ 28^{55}$
$\sqrt[55]{55!} \ \textless \ \sqrt[55]{28^{55}}$
$\sqrt[55]{2^{51} * 3^{26} * 5^{12} * 7^{8} * 11^{4} * 13^{4} * 17^{3} * 19^{2} * 23^{2} * 29 * 31 * 37 * 41 * 43 * 47 * 53} \ \textless \ \sqrt[55]{7^{55} * 2^{110}}$
$\sqrt[55]{3^{26} * 5^{12} * 11^{4} * 13^{4} * 17^{3} * 19^{2} * 23^{2} * 29 * 31 * 37 * 41 * 43 * 47 * 53} \ \textless \ \sqrt[55]{7^{47} * 2^{59}}$
$3^{26} * 5^{12} * 11^{4} * 13^{4} * 17^{3} * 19^{2} * 23^{2} * 29 * 31 * 37 * 41 * 43 \ \textless \ 7^{47} * 2^{59}$