(6-%E2%88%92-3i).jpeg "(6 + 3i)(6 − 3i)")
Multiplying Complex Numbers: (6 + 3i)(6 - 3i)
This article will explore the multiplication of the complex numbers (6 + 3i) and (6 - 3i). We'll use the distributive property and the fact that i² = -1 to simplify the expression and arrive at a real number result.
Understanding the Problem
We have two complex numbers:
- (6 + 3i): This is in the form of (a + bi), where 'a' and 'b' are real numbers, and 'i' is the imaginary unit.
- (6 - 3i): This is the complex conjugate of (6 + 3i).
The product of a complex number and its conjugate always results in a real number.
Solution
Let's multiply the complex numbers using the distributive property (FOIL method):
(6 + 3i)(6 - 3i) = 6(6 - 3i) + 3i(6 - 3i)
Expanding the expression:
= 36 - 18i + 18i - 9i²
Since i² = -1:
= 36 - 9(-1)
= 36 + 9
= 45
Conclusion
Therefore, the product of (6 + 3i) and (6 - 3i) is 45. This result highlights the important property that the product of a complex number and its conjugate always yields a real number.