Let's solve the multiplication of the complex numbers [tex]\((6 + \sqrt{-64})(3 - \sqrt{-16})\)[/tex].
Step 1: Simplify the complex numbers.
[tex]\[
a = 6 + \sqrt{-64}
\][/tex]
Note that [tex]\(\sqrt{-64} = \sqrt{64 \cdot -1} = \sqrt{64} \cdot \sqrt{-1} = 8i\)[/tex].
So,
[tex]\[
a = 6 + 8i
\][/tex]
[tex]\[
b = 3 - \sqrt{-16}
\][/tex]
Note that [tex]\(\sqrt{-16} = \sqrt{16 \cdot -1} = \sqrt{16} \cdot \sqrt{-1} = 4i\)[/tex].
So,
[tex]\[
b = 3 - 4i
\][/tex]
Now, we need to multiply these two complex numbers:
[tex]\[
(6 + 8i)(3 - 4i)
\][/tex]
Step 2: Distribute each term in the first complex number by each term in the second complex number:
[tex]\[
(6 + 8i)(3 - 4i) = 6 \cdot 3 + 6 \cdot (-4i) + 8i \cdot 3 + 8i \cdot (-4i)
\][/tex]
Let's calculate each term one by one:
1. Calculating [tex]\(6 \cdot 3\)[/tex]:
[tex]\[
6 \cdot 3 = 18
\][/tex]
2. Calculating [tex]\(6 \cdot (-4i)\)[/tex]:
[tex]\[
6 \cdot (-4i) = -24i
\][/tex]
3. Calculating [tex]\(8i \cdot 3\)[/tex]:
[tex]\[
8i \cdot 3 = 24i
\][/tex]
4. Calculating [tex]\(8i \cdot (-4i)\)[/tex]:
[tex]\[
8i \cdot (-4i) = -32i^2
\][/tex]
Since [tex]\(i^2 = -1\)[/tex], we have:
[tex]\[
-32i^2 = -32 \cdot (-1) = 32
\][/tex]
Step 3: Adding all the terms together:
[tex]\[
18 - 24i + 24i + 32
\][/tex]
Step 4: Simplify the expression:
Combining real parts:
[tex]\[
18 + 32 = 50
\][/tex]
Combining imaginary parts:
[tex]\[
-24i + 24i = 0
\][/tex]
So, the final result is:
[tex]\[
50 + 0i = 50
\][/tex]
Thus, the multiplication [tex]\((6 + \sqrt{-64})(3 - \sqrt{-16})\)[/tex] simplifies to [tex]\(50\)[/tex].
