%2F(3-i)(1%2B2i).jpeg "(2+i)/(3-i)(1+2i)")
Simplifying Complex Expressions: A Step-by-Step Guide
This article will guide you through the process of simplifying the complex expression (2 + i) / (3 - i)(1 + 2i).
Understanding Complex Numbers
Complex numbers are numbers of the form a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit (where i² = -1). The standard way to simplify a complex expression is to express it in the form a + bi.
Simplifying the Expression
Let's break down the simplification process:
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Simplify the denominator:
- Multiply the two complex numbers in the denominator: (3 - i)(1 + 2i) = 3 + 6i - i - 2i² = 3 + 5i + 2 = 5 + 5i
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Rewrite the expression with the simplified denominator:
- Our expression now becomes (2 + i) / (5 + 5i).
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Multiply numerator and denominator by the complex conjugate of the denominator:
- The complex conjugate of 5 + 5i is 5 - 5i.
- Multiplying both numerator and denominator by 5 - 5i gives: [(2 + i) / (5 + 5i)] * [(5 - 5i) / (5 - 5i)] = [(2 + i)(5 - 5i)] / [(5 + 5i)(5 - 5i)]
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Expand the numerator and denominator:
- Numerator: (2 + i)(5 - 5i) = 10 - 10i + 5i - 5i² = 10 - 5i + 5 = 15 - 5i
- Denominator: (5 + 5i)(5 - 5i) = 25 - 25i² = 25 + 25 = 50
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Express the result in the form a + bi:
- Our simplified expression becomes (15 - 5i) / 50 = 3/10 - 1/10i.
Conclusion
Therefore, the simplified form of the complex expression (2 + i) / (3 - i)(1 + 2i) is 3/10 - 1/10i.
This step-by-step process demonstrates how to simplify complex expressions by using the concept of complex conjugates and basic arithmetic operations.