All look same

Algebra Level 2

Which among the following is the largest?
$\large 3^{2^{2^2}}, \quad [(3^2)^2]^2, \quad 3^{2 \times 2 \times 2}, \quad 3222$

3^{2 \times 2 \times 2}

All are equal.

[(3^2)^2]^2

\large 3^{2^{2^2}}

3222

1 solution

Hung Woei Neoh
Apr 18, 2016

$3222 = 3222$ (lol)

$[(3^2)^2]^2 = [9^2]^2 = 81^2 = 6561$ . This can also be written as $3^8$

$3^{2 \cdot 2 \cdot 2} = 3^8 = 6561$

$3^{2^{2^2}} = 3^{2^4} = 3^{16} = (3^8)^2 = 6561^2$

Therefore: $3222 < [(3^2)^2]^2 = 3^{2 \cdot 2 \cdot 2} < 3^{2^{2^2}}$

The largest number is $\boxed{3^{2^{2^2}}}$