%5E2.jpeg "(8+7i)^2")
Squaring Complex Numbers: (8 + 7i)²
This article will explore the process of squaring the complex number (8 + 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.
Squaring a Complex Number
To square a complex number, we simply multiply it by itself. In this case, we have:
(8 + 7i)² = (8 + 7i)(8 + 7i)
Expanding the Product
We can expand this product using the distributive property (FOIL method):
(8 + 7i)(8 + 7i) = 8(8) + 8(7i) + 7i(8) + 7i(7i)
Simplifying the terms:
= 64 + 56i + 56i + 49i²
Remembering i² = -1
Since i² is defined as -1, we can substitute:
= 64 + 56i + 56i - 49
Combining Real and Imaginary Terms
Finally, combining the real and imaginary terms:
= (64 - 49) + (56 + 56)i
= 15 + 112i
Conclusion
Therefore, (8 + 7i)² is equal to 15 + 112i. This demonstrates how to square a complex number and how to simplify the result by combining real and imaginary terms.