%5E2%3D(4)%5E2%2B(6i)%5E2.jpeg "(4+6i)^2=(4)^2+(6i)^2")
Debunking a Common Misconception: (4 + 6i)² ≠ 4² + (6i)²
It's a common mistake to assume that squaring a complex number like (4 + 6i) is as simple as squaring each term individually. However, this is incorrect. Let's break down why:
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, defined as the square root of -1.
The Correct Approach
To square a complex number, we need to remember that multiplication is distributive. Therefore, (4 + 6i)² is equivalent to:
(4 + 6i)² = (4 + 6i)(4 + 6i)
Now, we expand using the distributive property:
(4 + 6i)(4 + 6i) = 4 * 4 + 4 * 6i + 6i * 4 + 6i * 6i
Simplifying the terms:
16 + 24i + 24i + 36i²
Remember that i² = -1. Substituting this:
16 + 24i + 24i - 36
Combining like terms:
-20 + 48i
The Takeaway
The correct answer is (4 + 6i)² = -20 + 48i. It's important to understand that complex numbers follow different rules for multiplication than real numbers. Simply squaring each term individually leads to an incorrect result. Always remember to apply the distributive property and the fact that i² = -1 when working with complex numbers.