%5E2.jpeg "(1-4i)^2")
Simplifying Complex Numbers: (1-4i)^2
In mathematics, complex numbers are expressed in the form a + bi, where 'a' and 'b' are real numbers and 'i' is the imaginary unit, defined as √-1. To simplify the expression (1-4i)^2, we need to expand it using the distributive property or the FOIL method.
Expanding the Expression
Using the FOIL method, we multiply each term in the first binomial by each term in the second binomial:
(1 - 4i) * (1 - 4i) = (1 * 1) + (1 * -4i) + (-4i * 1) + (-4i * -4i)
Simplifying further:
= 1 - 4i - 4i + 16i^2
Since i^2 = -1, we can substitute:
= 1 - 4i - 4i + 16(-1)
Combining like terms:
= 1 - 8i - 16
= -15 - 8i
Final Result
Therefore, the simplified form of (1-4i)^2 is -15 - 8i.