Answer :
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].
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].
