The exponent of 2 in the prime factorisation of 144, is

Using the factor tree for prime factorization, we have:

Therefore,

$144=2 \times 2 \times 2 \times 2 \times 3 \times 3$

$144=2^{4} \times 3^{2}$

Thus, the exponent of 2 in 144 is 4.

Hence the correct choice is (a).