Explanation:
you understand how exponents work, right ?
an exponent defines how often its base has to be multiplied by itself.
so,
(-1 + 3i)⁵
means that
(-1 + 3i)
is multiplied 5 times by itself.
and that means we can split this into sub-multiplication groups like
(-1 + 3i)²×(-1 + 3i)²×(-1 + 3i) = (-1 + 3i)⁴×(-1 + 3i)
the advantage is that squares are easily calculated.
(a + b)² = a² + 2ab + b²
so,
(-1 + 3i)² = (-1)² + 2×-1×3i + (3i)² = 1 - 6i - 9 = -8 - 6i
(-1 + 3i)²×(-1 + 3i)² = ((-1 + 3i)²)² = (-8 - 6i)² =
= (-8)² + 2×-8×-6i + (-6i)² = 64 + 96i - 36 = 28 + 96i
(-1 + 3i)⁵ = (28 + 96i)(-1 + 3i) = -28 + 84i - 96i - 288 =
= -316 - 12i