I am

$\large { i }^{ 1872 } \\ \large = { \left( { i }^{ 4 } \right) }^{ 468 } \\ \large = { \left( 1 \right) }^{ 468 } \\ \large = \color{#20A900}{\boxed{1}}$

The definition of i is that ${i}^{2}={-1}$
${i}^{1}$ = i
${i}^{3}={i}^{2}*{i}^{1}={i}^{3}={-i}$
${i}^{4}={i}^{2}*{i}^{2}={1}$
Then, it repeats. If you divide the exponent by 4, if you get remainder 1, its i. (1 1 1*i)
If you get remainder 2, its -1.
If you get remainder 3, its -i. If you get remainder 0, its 1. 1872 mod 4 is 0, so the answer is 1.