%5E2.jpeg "(4+9i)^2")
Squaring Complex Numbers: (4 + 9i)^2
This article explores the process of squaring the complex number (4 + 9i).
Understanding Complex Numbers
Complex numbers are numbers that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, defined as the square root of -1 (i.e., i² = -1).
Squaring (4 + 9i)
To square a complex number, we simply multiply it by itself.
(4 + 9i)² = (4 + 9i) * (4 + 9i)
We can use the distributive property (or FOIL method) to expand this expression:
(4 + 9i) * (4 + 9i) = 4 * 4 + 4 * 9i + 9i * 4 + 9i * 9i
Simplifying the terms:
- 16 + 36i + 36i + 81i²
Since i² = -1:
- 16 + 36i + 36i - 81
Combining the real and imaginary terms:
- (16 - 81) + (36 + 36)i
Therefore, (4 + 9i)² = -65 + 72i
Conclusion
The square of the complex number (4 + 9i) is -65 + 72i. This process demonstrates the fundamental operations involved in working with complex numbers. Understanding these operations is crucial for solving various mathematical problems involving complex numbers.