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Simplifying Complex Numbers: (3 + 6i)² / 2i
This article will guide you through the process of simplifying the complex number expression (3 + 6i)² / 2i.
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
Let's break down the simplification process step-by-step:
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Expand the Square: First, we need to expand the square of the complex number in the numerator: (3 + 6i)² = (3 + 6i)(3 + 6i) = 9 + 18i + 18i + 36i² Using the fact that i² = -1, we can simplify: 9 + 18i + 18i + 36i² = 9 + 36i - 36 = -27 + 36i
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Divide by 2i: Now we need to divide the simplified numerator (-27 + 36i) by the denominator 2i. To divide by a complex number, we multiply both numerator and denominator by the complex conjugate of the denominator: (-27 + 36i) / 2i * (-2i) / (-2i) = (54i + 72i²) / -4i²
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Simplify: Using i² = -1, we can further simplify: (54i + 72i²) / -4i² = (54i - 72) / 4 = -18 + 13.5i
Therefore, the simplified form of the complex number expression (3 + 6i)² / 2i is -18 + 13.5i.
Conclusion
By following the steps outlined above, we successfully simplified the complex number expression. Remember that understanding the properties of complex numbers, particularly the imaginary unit (i), is crucial for working with such expressions.