%5E2.jpeg "(7-2i)^2")
Squaring Complex Numbers: (7-2i)²
This article will walk through the process of squaring the complex number (7-2i).
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.
Squaring (7-2i)
To square (7-2i), we simply multiply it by itself:
(7-2i)² = (7-2i)(7-2i)
We can use the FOIL method (First, Outer, Inner, Last) to expand the product:
- First: 7 * 7 = 49
- Outer: 7 * -2i = -14i
- Inner: -2i * 7 = -14i
- Last: -2i * -2i = 4i²
Combining the terms, we get:
49 - 14i - 14i + 4i²
Remember that i² = -1. Substituting this into our expression:
49 - 14i - 14i + 4(-1)
Simplifying:
49 - 14i - 14i - 4
Combining real and imaginary terms:
(49 - 4) + (-14 - 14)i
Finally, we arrive at the solution:
(7-2i)² = 45 - 28i
Conclusion
Squaring a complex number involves expanding the expression and simplifying the result, using the knowledge that i² = -1. The result of (7-2i)² is 45 - 28i.