To simplify the expression \((6 - 3i)(3 - 5i)\) using the Distributive Property (also known as the FOIL method for binomials), we will multiply each term in the first binomial by each term in the second binomial.
- First terms: \(6 \cdot 3 = 18\)
- Outer terms: \(6 \cdot (-5i) = -30i\)
- Inner terms: \(-3i \cdot 3 = -9i\)
- Last terms: \(-3i \cdot (-5i) = 15i^2\)
Now, combine these results:
\[ 18 - 30i - 9i + 15i^2 \]
Next, recall that \(i^2 = -1\). Thus, we can substitute \(-1\) for \(i^2\):
\[ 15i^2 = 15(-1) = -15 \]
Now, substitute this back into the expression:
\[ 18 - 30i - 9i - 15 \]
Combine the real parts and the imaginary parts:
\[ (18 - 15) + (-30i - 9i) = 3 - 39i \]
Therefore, the simplified form of \((6 - 3i)(3 - 5i)\) is:
\[ \boxed{3 - 39i} \]
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