(2-3i%2F3-4i).jpeg "(2+3i/3+4i)(2-3i/3-4i)")
Simplifying Complex Fractions
This article will guide you through the process of simplifying the complex fraction:
(2 + 3i / 3 + 4i) * (2 - 3i / 3 - 4i)
Understanding Complex Numbers
Before we dive into the simplification, let's understand the basics of complex numbers. Complex numbers are numbers of the form a + bi, where a and b are real numbers, and i is the imaginary unit, defined as the square root of -1 (i.e., i² = -1).
Simplifying the Fractions
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Focus on the Individual Fractions: We start by simplifying the individual fractions within the parentheses. To do this, we multiply both the numerator and denominator of each fraction by the complex conjugate of the denominator. This eliminates the imaginary term in the denominator.
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Fraction 1: (2 + 3i) / (3 + 4i)
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Complex Conjugate of (3 + 4i) is (3 - 4i)
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Multiply numerator and denominator by (3 - 4i):
- Numerator: (2 + 3i) * (3 - 4i) = 6 - 8i + 9i - 12i² = 18 + i
- Denominator: (3 + 4i) * (3 - 4i) = 9 - 16i² = 25
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Simplified Fraction 1: (18 + i) / 25
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Fraction 2: (2 - 3i) / (3 - 4i)
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Complex Conjugate of (3 - 4i) is (3 + 4i)
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Multiply numerator and denominator by (3 + 4i):
- Numerator: (2 - 3i) * (3 + 4i) = 6 + 8i - 9i - 12i² = 18 - i
- Denominator: (3 - 4i) * (3 + 4i) = 9 - 16i² = 25
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Simplified Fraction 2: (18 - i) / 25
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Multiplying the Simplified Fractions: Now that we have simplified the individual fractions, we multiply the results:
- ((18 + i) / 25) * ((18 - i) / 25)
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Expanding: Multiply the numerators and denominators:
- Numerator: (18 + i) * (18 - i) = 324 - i² = 325
- Denominator: 25 * 25 = 625
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Final Result: The simplified form of the complex fraction is:
- (325 / 625) which can be further simplified to 13 / 25.
Conclusion
By understanding the basics of complex numbers and using the concept of complex conjugates, we successfully simplified the complex fraction. The final result, 13 / 25, is a real number.