Question

Arrange the following in ascending order :
$2^{5555}, 3^{3333}, 6^{2222}$

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Answer 1

$\Huge3^{3333}<2^{5555}<6^{2222}$

$\Huge\text{\underline{\underline{Idea}}}$

The main idea to solve this equation is the laws of exponents. As we see here, there is a common factor in the exponent, 1111.

To compare these numbers, we can divide numbers by each other.

$\huge\text{\underline{\underline{Explanation}}}$

$\cdots\longrightarrow\dfrac{2^{5555}}{3^{3333}}=\left(\dfrac{32}{27}\right)^{1111}>1$

$\cdots\longrightarrow\boxed{2^{5555}>3^{3333}}$

$\cdots\longrightarrow\dfrac{3^{3333}}{6^{2222}}=\left(\dfrac{27}{36}\right)^{1111}<1$

$\cdots\longrightarrow\boxed{3^{3333}<6^{2222}}$

$\cdots\longrightarrow\dfrac{6^{2222}}{2^{5555}}=\left(\dfrac{36}{32}\right)^{1111}>1$

$\cdots\longrightarrow\boxed{6^{2222}>2^{5555}}$

$\huge\text{\underline{\underline{Final answer}}}$

Required answer.

$\cdots\longrightarrow\boxed{3^{3333}<2^{5555}<6^{2222}}$