%5E2.jpeg "(2-7i)^2")
Squaring a Complex Number: (2 - 7i)^2
This article will explore the process of squaring the complex number (2 - 7i).
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.e., i^2 = -1).
Squaring (2 - 7i)
To square a complex number, we simply multiply it by itself:
(2 - 7i)^2 = (2 - 7i) * (2 - 7i)
We can expand this using the distributive property (or FOIL method):
- (2 * 2) + (2 * -7i) + (-7i * 2) + (-7i * -7i)
Simplifying:
- 4 - 14i - 14i + 49i^2
Since i^2 = -1, we can substitute and further simplify:
- 4 - 14i - 14i + 49(-1)
- 4 - 28i - 49
- -45 - 28i
Therefore, (2 - 7i)^2 = -45 - 28i.
Conclusion
Squaring a complex number involves multiplying it by itself and then simplifying the result using the properties of imaginary numbers. In this case, we found that (2 - 7i)^2 is equal to -45 - 28i.