Simplifying Complex Expressions: (6i)(77i) + (i)(1+3i)
This article will guide you through the process of simplifying the complex expression (6i)(77i) + (i)(1+3i).
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 (i² = 1).
Simplifying the Expression

Distribute: First, we distribute the terms in each set of parentheses:
(6i)(77i) + (i)(1+3i) = 42i  42i² + i + 3i²

Substitute i² = 1: Next, we substitute 1 for every instance of i²:
42i  42i² + i + 3i² = 42i  42(1) + i + 3(1)

Combine Real and Imaginary Terms: Combine the real terms (those without 'i') and the imaginary terms (those with 'i'):
42i + 42 + i  3 = (42  3) + (42 + 1)i

Simplify: Finally, perform the arithmetic to get the simplified expression:
(42  3) + (42 + 1)i = 39 + 43i
Conclusion
Therefore, the simplified form of the complex expression (6i)(77i) + (i)(1+3i) is 39 + 43i.
This demonstrates the key steps involved in simplifying complex expressions: distribution, substitution of i², and combining real and imaginary terms.