Question

Please help. see photo below

Answer 1

48 + 2i

Explanation:

We are given:

`\displaystyle{3(5-1)^2 + √(-36) + 4i^3}\\\\\displaystyle{3(4)^2 + √(-36) + 4i^3}\\\\\displaystyle{3\cdot 16 + √(-36) + 4i^3}\\\\\displaystyle{48+ √(-36) + 4i^3}`

Recall that:

`\displaystyle{√(-a) = √(a){√(-1) \ \, \ (a > 0)}`

Therefore:

`\displaystyle{48 + √(-36) + 4i^3}\\\\\displaystyle{48+ √(36)√(-1) + 4i^3}`

Also recall the definition of imaginary number that:

`\displaystyle{√(-1) = i}`

Hence:

`\displaystyle{48 + √(36)i + 4i^3}\\\\\displaystyle{48 + 6i + 4i^3}`

We know that:

`\displaystyle{i^3 = i \cdot i \cdot i}\\\\\displaystyle{i^3 = i^2 \cdot i}\\\\\displaystyle{i^3 = -i}`

Note that:

`\displaystyle{i^2 = -1}`

Therefore:

`\displaystyle{48 + 6i + 4\cdot (-i)}\\\\\displaystyle{48 + 6i - 4i}\\\\\displaystyle{48 + 2i}`

Therefore, the answer is 48 + 2i.