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# Nothing out of order?

Help me arrange $$180^{4}$$,$$90^{8}$$,$$60^{12}$$,$$40^{18}$$,$$30^{24}$$,$$15^{48}$$,$$10^{72}$$,$$5^{144}$$,$$4^{180}$$,$$3^{240}$$,$$80^{9}$$,$$72^{10}$$ in $\color{BLUE}{\huge{ascending }}$order.

Note by Bryan Lee Shi Yang
1 year, 10 months ago

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Take the $$720$$ th root of all numbers. Then, all the terms come into the form of $$x^{1/x}$$. Since it is a decreasing function for $$x > e$$, if $$y > x > e$$, then $$x^{1/x} > y^{1/y}$$. Therefore, the order is $$3^{1/3} > 4^{1/4}> ... > 180^{1/180}$$. Raising each term to the $$720$$th power, $$3^{240} > 4^{180} > ... > 180^4$$ · 1 year, 10 months ago