Question

Can someone please help me on this one!!!!

Simplify each expression using the definition, identities, and properties of imaginary numbers. Match each term in the list on the left to its equivalent simplified form on the right.

1. (i3)2(-i0)3(3i2)^4 81
2. (i3i-5)2(i4i-3)^0 -i
3. (-i4i2)(2i)2(i-1ii5)3(i0)^-3 1
4. i3[(i2i3i4)(i0i2i5)]^2 -4

Answer 1

This appears to be about rules of exponents as much as anything. The applicable "definitions, identities, and properties" are

i^0 = 1 . . . . . as is true for any non-zero value to the zero power

i^1 = i . . . . . . as is true for any value to the first power

i^2 = -1 . . . . . from the definition of i

i^3 = -i . . . . . = (i^2)·(i^1) = -1·i = -i

i^n = i^(n mod 4) . . . . . where "n mod 4" is the remainder after division by 4

1. = -3^4·i^(3·2+0+2·4) = -81·i^14 = 81

2. = i^((3-5)·2+0 = i^-4 = 1

3. = -2^2·i^(4+2+2+(-1+1+5)·3+0) = -4·i^23 = 4i

4. = i^(3+(2+3+4+0+2+5)·2) = i^35 = -i

Explanation:

Answer 2

This appears to be about rules of exponents as much as anything. The applicable "definitions, identities, and properties" are
i^0 = 1 . . . . . as is true for any non-zero value to the zero power
i^1 = i . . . . . . as is true for any value to the first power
i^2 = -1 . . . . . from the definition of i
i^3 = -i . . . . . = (i^2)·(i^1) = -1·i = -i
i^n = i^(n mod 4) . . . . . where "n mod 4" is the remainder after division by 4


1. = -3^4·i^(3·2+0+2·4) = -81·i^14 = 81

2. = i^((3-5)·2+0 = i^-4 = 1

3. = -2^2·i^(4+2+2+(-1+1+5)·3+0) = -4·i^23 = 4i

4. = i^(3+(2+3+4+0+2+5)·2) = i^35 = -i