%5E3.jpeg "(1-2i)^3")
Simplifying (1-2i)³
This article will guide you through the process of simplifying the complex number expression (1-2i)³.
Understanding Complex Numbers
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).
Simplifying the Expression
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Expansion: We can expand the expression using the distributive property: (1 - 2i)³ = (1 - 2i) * (1 - 2i) * (1 - 2i)
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Multiplying: Let's first multiply the first two terms: (1 - 2i) * (1 - 2i) = 1 - 2i - 2i + 4i² Substituting i² = -1, we get: 1 - 2i - 2i + 4(-1) = -3 - 4i
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Final Multiplication: Now we multiply the result by (1 - 2i): (-3 - 4i) * (1 - 2i) = -3 + 6i - 4i + 8i² Again, substituting i² = -1: -3 + 6i - 4i + 8(-1) = -11 + 2i
Final Result
Therefore, (1 - 2i)³ simplifies to -11 + 2i.
Visualizing Complex Numbers
Complex numbers can be visualized on a complex plane, where the horizontal axis represents the real part and the vertical axis represents the imaginary part. The result -11 + 2i would be represented as a point 11 units to the left of the origin and 2 units above.