Question

question 3 (essay worth 10 points)
(01.07r mc)

show all work to multiply \\((3 + \sqrt{-16})(6 - \sqrt{-64})\\).

Explanation:

Simplify the radical expressions

Using the Imaginary Unit and Complex Numbers knowledge points:

$$ \sqrt{-16} = \sqrt{16} \cdot \sqrt{-1} = 4i $$

$$ \sqrt{-64} = \sqrt{64} \cdot \sqrt{-1} = 8i $$

Rewrite the expression

Substitute the simplified imaginary terms back into the original product:

$$ (3 + \sqrt{-16})(6 - \sqrt{-64}) = (3 + 4i)(6 - 8i) $$

Expand the product

Using the Multiplying Complex Numbers knowledge point, apply the FOIL method to expand the binomials:

$$ (3 + 4i)(6 - 8i) = 3(6) + 3(-8i) + 4i(6) + 4i(-8i) $$

$$ = 18 - 24i + 24i - 32i^2 $$

Simplify using the imaginary unit identity

Using the Imaginary Unit identity \(i^2 = -1\):

$$ 18 - 24i + 24i - 32(-1) $$

$$ = 18 - 24i + 24i + 32 $$

Combine like terms

Combine the real parts and the imaginary parts:

$$ (18 + 32) + (-24i + 24i) = 50 + 0i = 50 $$

Answer:

\(50\)