Only i^19 and i^71 are equivalent to -i as their exponents, when divided by 4, leave a remainder of 3, which corresponds to the value of -i.
The complex number -i is equivalent to some powers of i. Since i has a repeating pattern every fourth power, that is i1 = i, i2 = -1, i3 = -i, and i4 = 1, we can determine the equivalents by dividing the exponent by 4 and looking at the remainder.
For exponents that have a remainder of 0 when divided by 4, i to that power equals 1. If the remainder is 1, it's equal to i. If the remainder is 2, it's equal to -1, and if the remainder is 3, it's equal to -i. Hence, the equivalents of -i are the powers of i where the exponent, when divided by 4, leaves a remainder of 3. These are i3, i7, i11, and so on, where the exponent is of the form 4n+3.
From the options given:
- i25 (4*6+1) is equivalent to i.
- i57 (4*14+1) is equivalent to i.
- i19 (4*4+3) is equivalent to -i.
- i64 (4*16) is equivalent to 1.
- i14 (4*3+2) is equivalent to -1.
- i42 (4*10+2) is equivalent to -1.
- i37 (4*9+1) is equivalent to i.
- i71 (4*17+3) is equivalent to -i.
Therefore, the values equivalent to -i are i19 and i71.