text: what is the square root of the square root of -16 in imaginary numbers? -16 as in negative 16... palindrome geoff is right about how to arrive at the root he described, however, that is not the whole story.  just as there are two numbers that satisfy the meaning of √16 (that is, 4 and -4) there are actually four numbers that satisfy the meaning of √(√(-16)): ● √2 + i√2 ● -√2 + i√2 ● √2 - i√2 ● -√2 - i√2  geoff's analysis about "turning the angle halfway" toward the positive x axis is a good intuitive way of understanding what is going on. however, what he left out is that it doesn't matter which direction you turn, clockwise or counter-clockwise. both directions are equally valid and both should be taken into account.  for √(-16), turning clockwise gives you 4i, counter-clockwise gives you -4i  for √(4i), turning clockwise gives you √2 + i√2, counter-clockwise gives you -√2 - i√2 (remember, you are going halfway to the positive x axis the long way)  for √(-4i), turning clockwise gives you -√2 + i√2, counter-clockwise gives you √2 - i√2