(3%2B4%2Fi)(5%2Bi)%5E-1.jpeg "(1+2/i)(3+4/i)(5+i)^-1")
Simplifying Complex Expressions: (1+2/i)(3+4/i)(5+i)^-1
This article will guide you through the process of simplifying the complex expression (1+2/i)(3+4/i)(5+i)^-1. We'll use fundamental concepts of complex numbers and algebraic manipulation to achieve the solution.
Understanding Complex Numbers
First, let's refresh our understanding of complex numbers. A complex number is a number of the form a + bi, where a and b are real numbers and i is the imaginary unit, defined as √-1.
Simplifying the Expression
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Simplifying the Denominator:
We start by simplifying the term (5+i)^-1. Remember that x^-1 = 1/x. Therefore,
(5+i)^-1 = 1/(5+i). To get rid of the complex number in the denominator, we multiply both numerator and denominator by the complex conjugate of (5+i), which is (5-i): 1/(5+i) * (5-i)/(5-i) = (5-i)/(25 - i^2) = (5-i)/26. -
Simplifying the First Two Factors: Now we simplify (1+2/i)(3+4/i). Remember that 1/i = -i. Therefore: (1+2/i)(3+4/i) = (1-2i)(3-4i) = 3 - 4i - 6i + 8i^2 = -5 - 10i
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Combining the Simplified Terms: Finally, we combine the simplified terms: (-5 - 10i) * (5-i)/26 = (-25 + 5i - 50i + 10i^2)/26 = (-35 - 45i)/26
The Final Solution
Therefore, the simplified form of (1+2/i)(3+4/i)(5+i)^-1 is (-35 - 45i)/26.
Conclusion
Simplifying complex expressions often involves utilizing the properties of complex numbers and algebraic manipulation techniques. By following the steps outlined above, we successfully simplified the given expression to a more manageable form.