Imaginary numbers are typically simplified in the same way that expressions with the variable x are simplified, with the exception of the expression i^2, which has a value equal to -1. For example, the expression 2i + 8i can be simplified by writing it first as (2 + 8) * i, with the final expression written as 10i.

For division problems, imaginary numbers are simplified by clearing the denominator by multiplying the expression by the conjugate of the denominator over itself, which is the same as multiplying by one and does not change the value of the expression. For example, in the problem (2 + i)/(3 + 4i) is simplified by multiplying it by (3 - 4i)/(3 - 4i), which results in the simplified fraction (10 - i)/25.

Any time i is taken to an odd exponent, it is equal to either i or -i. When taken to an even exponent, i is always equal to -1. When i is expressed to the power of zero, it is equal to 1. In the problem 4i * 5i, the simplified form comes from rewriting the expression as (4 * 5)(i^2). Since 4 * 5 is equal to 20 and i^2 is equal to -1, the final solution for the problem is -12.