%5E2.jpeg "(5+i)^2")
Squaring a Complex Number: (5 + i)^2
This article will explore the process of squaring the complex number (5 + i).
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., i² = -1).
Squaring the Complex Number
To square (5 + i), we simply multiply it by itself:
(5 + i)² = (5 + i)(5 + i)
We can expand this using the distributive property (or FOIL method):
(5 + i)(5 + i) = 5 * 5 + 5 * i + i * 5 + i * i
Simplifying:
= 25 + 5i + 5i + i²
Since i² = -1, we can substitute:
= 25 + 5i + 5i - 1
Combining real and imaginary terms:
= (25 - 1) + (5 + 5)i
= 24 + 10i
The Result
Therefore, the square of (5 + i) is 24 + 10i.
This demonstrates how squaring a complex number involves multiplying it by itself and utilizing the property that i² = -1 to simplify the expression.