Balancing act

Algebra Level 3

Which of the following is largest?

$\huge 3^{5^{3^{3}}}, \quad 5^{3^{3^{3}}}, \quad 3^{3^{5^{3}}}, \quad3^{3^{3^{5}}}$

\Large 3^{3^{5^{3}}}

\Large 3^{5^{3^{3}}}

\Large 5^{3^{3^{3}}}

\Large 3^{3^{3^{5}}}

1 solution

Chew-Seong Cheong
Oct 1, 2018

Since $5^3 = 125$ and $3^5 = 243$ , $\implies 3^5 > 5^3$ . Therefore, $3^{3^{\color{#D61F06}3^5}} > 3^{3^{\color{#D61F06}5^3}}$ and ${\color{#D61F06}3}^{{\color{#D61F06}5}^{3^3}} > {\color{#D61F06}5}^{{\color{#D61F06}3}^{3^3}}.$ Now comparing $3^{3^5} = 3^{243}$ and $5^{3^3} = 5^{27} < 9^{27} = 3^{54} < 3^{243} = 3^{3^5}$ . Therefore, $3^{\color{#D61F06}3^{3^5}} > 3^{\color{#D61F06}5^{3^5}}$ and $\boxed{3^{3^{3^5}}}$ is the greatest .

Balancing act

1 solution

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