2.jpeg "(4 + 5i)2")
Expanding (4 + 5i)<sup>2</sup>
In mathematics, particularly in the realm of complex numbers, we often encounter expressions involving the square of a complex number. One such expression is (4 + 5i)<sup>2</sup>. Let's explore how to expand and simplify this expression.
Understanding Complex Numbers
A complex number is a number 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<sup>2</sup> = -1).
Expanding the Expression
To expand (4 + 5i)<sup>2</sup>, we can use the FOIL method (First, Outer, Inner, Last) or simply distribute the expression:
(4 + 5i)<sup>2</sup> = (4 + 5i)(4 + 5i)
= 4(4 + 5i) + 5i(4 + 5i)
= 16 + 20i + 20i + 25i<sup>2</sup>
Simplifying the Expression
Now, recall that i<sup>2</sup> = -1. Substituting this into our expression:
= 16 + 20i + 20i + 25(-1)
= 16 + 20i + 20i - 25
= -9 + 40i
Conclusion
Therefore, the expanded and simplified form of (4 + 5i)<sup>2</sup> is -9 + 40i. This demonstrates that squaring a complex number results in another complex number, where both the real and imaginary components can change.