(6i)(7-7i)+(i)(1+3i)

Jasonbradley

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Jun 16, 2024

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Simplifying Complex Expressions: (6i)(7-7i) + (i)(1+3i)

This article will guide you through the process of simplifying the complex expression (6i)(7-7i) + (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

1. Distribute: First, we distribute the terms in each set of parentheses:

   (6i)(7-7i) + (i)(1+3i) = 42i - 42i² + i + 3i²

2. Substitute i² = -1: Next, we substitute -1 for every instance of i²:

   42i - 42i² + i + 3i² = 42i - 42(-1) + i + 3(-1)

3. 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

4. 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)(7-7i) + (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.