%5E2.jpeg "(1-7i)^2")
Squaring Complex Numbers: (1-7i)^2
This article explores how to square the complex number (1-7i).
Understanding Complex Numbers
Complex numbers are numbers that consist of two parts: a real part and an imaginary part. They are typically written in the form a + bi, where:
- a is the real part
- b is the imaginary part
- i is the imaginary unit, defined as the square root of -1 (i.e., i² = -1)
Squaring (1-7i)
To square (1-7i), we simply multiply it by itself:
(1-7i)² = (1-7i)(1-7i)
To multiply these complex numbers, we use the distributive property (also known as FOIL):
(1-7i)(1-7i) = 1(1) + 1(-7i) - 7i(1) - 7i(-7i)
Now, let's simplify:
- 1(1) = 1
- 1(-7i) = -7i
- -7i(1) = -7i
- -7i(-7i) = 49i²
Since i² = -1, we can substitute:
49i² = 49(-1) = -49
Combining all the terms:
1 - 7i - 7i - 49 = -48 - 14i
Therefore, (1-7i)² = -48 - 14i.
Conclusion
Squaring complex numbers involves applying the distributive property and remembering that i² = -1. The result of squaring (1-7i) is the complex number -48 - 14i.