As a complex number, what is the product of (i^3 - i^2)(3i^4 + i)?

To find the product of the expression (i^3 - i^2)(3i^4 + i), we first need to simplify each part.

Starting with i^3 and i^2, we know the powers of the imaginary unit i follow a pattern:

• i^1 = i

• i^2 = -1

• i^3 = -i

• i^4 = 1

Using this, we can simplify i^3 - i^2:

• i^3 = -i

• i^2 = -1

Thus:

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

Next, we simplify 3i^4 + i:

• i^4 = 1 (as established), so:

3i^4 = 3 * 1 = 3.

Therefore:

3i^4 + i = 3 + i.

Now we can multiply the two expressions we simplified:

(1 - i)(3 + i).

Using the distributive property (also known as the FOIL method for binomials), we calculate:

1 * 3 = 3,

1 * i = i,