%5E2.jpeg "(3+5i)^2")
Squaring Complex Numbers: A Look at (3+5i)^2
In the world of complex numbers, squaring a number like (3+5i) might seem daunting, but it's actually quite straightforward. Let's break down the process step-by-step:
Understanding Complex Numbers
A complex number is a number of the form a + bi, where a and b are real numbers, and i is the imaginary unit, defined as the square root of -1.
Expanding the Square
To square (3+5i), we simply multiply it by itself:
(3 + 5i)² = (3 + 5i)(3 + 5i)
Now, we can use the distributive property (or FOIL method) to expand this expression:
(3 + 5i)(3 + 5i) = 3(3) + 3(5i) + 5i(3) + 5i(5i)
Simplifying the Expression
Let's simplify each term:
- 3(3) = 9
- 3(5i) = 15i
- 5i(3) = 15i
- 5i(5i) = 25i²
Remember that i² = -1. Substituting this into our expression:
9 + 15i + 15i + 25(-1)
Combining Like Terms
Now, combine the real and imaginary terms:
(9 - 25) + (15 + 15)i
This simplifies to:
-16 + 30i
Conclusion
Therefore, (3 + 5i)² is equal to -16 + 30i. By understanding the properties of complex numbers and using the distributive property, we can easily square complex numbers and obtain a new complex number as a result.