%5E2.jpeg "(7+3i)^2")
Expanding (7 + 3i)^2
This article explores the process of expanding the expression (7 + 3i)^2, where 'i' represents the imaginary unit (√-1).
Understanding Complex Numbers
Complex numbers are expressed in the form a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit.
Expanding the Expression
We can expand (7 + 3i)^2 using the distributive property or by recognizing it as a square of a binomial:
Method 1: Distributive Property
(7 + 3i)^2 = (7 + 3i)(7 + 3i)
Expanding using the distributive property:
= 7(7 + 3i) + 3i(7 + 3i) = 49 + 21i + 21i + 9i^2
Since i^2 = -1, we can substitute:
= 49 + 42i - 9 = 40 + 42i
Method 2: Square of a Binomial
(7 + 3i)^2 = (7)^2 + 2(7)(3i) + (3i)^2
Simplifying:
= 49 + 42i + 9i^2 = 49 + 42i - 9 = 40 + 42i
Conclusion
Therefore, the expanded form of (7 + 3i)^2 is 40 + 42i. This process demonstrates how to manipulate complex numbers and simplify expressions involving the imaginary unit.