(1%2B3i)%E2%88%92(1%2B3zi)(5%2B3i).jpeg "(1+3zi)(1+3i)−(1+3zi)(5+3i)")
Simplifying Complex Expressions
This article will guide you through simplifying the complex expression:
(1 + 3zi)(1 + 3i) - (1 + 3zi)(5 + 3i)
Understanding the Basics
Before we dive into the simplification, let's recall some key concepts about complex numbers:
- Complex Number: 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 (i² = -1).
- Multiplication of Complex Numbers: To multiply two complex numbers, we use the distributive property (FOIL method) just like we do with binomials.
Simplifying the Expression
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Distribute: First, we distribute the terms within each set of parentheses:
(1 + 3zi)(1 + 3i) - (1 + 3zi)(5 + 3i) = (1 + 3i + 3zi + 9zi²) - (5 + 3i + 15zi + 9zi²)
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Simplify the Imaginary Terms: Replace i² with -1 and combine like terms:
(1 + 3i + 3zi - 9) - (5 + 3i + 15zi - 9) = (-8 + 3i + 3zi) - (-4 + 3i + 15zi)
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Combine Real and Imaginary Components: Group the real and imaginary terms separately:
(-8 + 4) + (3 - 3)i + (3 - 15)zi = -4 - 12zi
Final Result
Therefore, the simplified form of the expression (1 + 3zi)(1 + 3i) - (1 + 3zi)(5 + 3i) is -4 - 12zi.
Conclusion
By carefully applying the distributive property and simplifying the imaginary terms, we successfully simplified the complex expression. Remember to combine the real and imaginary components separately for the final result.