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.

#NumberTheory

Easy Math Editor

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
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`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
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Comments

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, $3^{240} > 4^{180} > ... > 180^4$

Siddhartha Srivastava - 6 years, 3 months ago

First take LCM of all the powers and then bring all the numbers in the power equal to LCM which is 720. Then the arrangement becomes 3^240,4^180,5^144,10^72,15^48, 30^24,40^18,60^12,72^10,80^9,90^8,180^4

Aditya Chauhan - 6 years, 2 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$