%5E2.jpeg "(-1+3i)^2")
Squaring Complex Numbers: A Step-by-Step Guide for (-1+3i)^2
This article will guide you through the process of squaring the complex number (-1+3i). We'll break down the steps and explain the concepts involved.
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 the Complex Number
To square (-1 + 3i), we simply multiply it by itself:
(-1 + 3i)^2 = (-1 + 3i) * (-1 + 3i)
Expanding the Product
We can use the distributive property (or FOIL method) to expand the product:
(-1 + 3i) * (-1 + 3i) = (-1)(-1) + (-1)(3i) + (3i)(-1) + (3i)(3i)
Simplifying the Expression
Now we simplify the expression:
1 - 3i - 3i + 9i^2
Remember that i^2 = -1, so we substitute:
1 - 3i - 3i - 9
Combining Real and Imaginary Terms
Finally, we combine the real and imaginary terms:
(1 - 9) + (-3 - 3)i = -8 - 6i
Conclusion
Therefore, (-1 + 3i)^2 = -8 - 6i. We have successfully squared the complex number by expanding the product, simplifying using the definition of i^2, and combining real and imaginary terms.