%5E3.jpeg "(-2-1/3i)^3")
Simplifying Complex Numbers: (-2 - 1/3i)³
This article will guide you through the process of simplifying the complex number (-2 - 1/3i)³. We'll break down the problem into manageable steps, highlighting the key concepts involved.
Understanding Complex Numbers
Complex numbers are numbers that can be expressed in the form a + bi, where:
- a and b are real numbers
- i is the imaginary unit, defined as the square root of -1 (i² = -1)
Simplifying the Expression
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Expand the expression: We need to cube the complex number, meaning we multiply it by itself three times. (-2 - 1/3i)³ = (-2 - 1/3i) * (-2 - 1/3i) * (-2 - 1/3i)
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Multiply the first two terms: (-2 - 1/3i) * (-2 - 1/3i) = 4 + 2/3i + 2/3i + 1/9i² = 4 + 4/3i - 1/9 (since i² = -1) = 35/9 + 4/3i
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Multiply the result by the remaining term: (35/9 + 4/3i) * (-2 - 1/3i) = -70/9 - 10/9i - 8/3i - 4/9i² = -70/9 - 34/9i + 4/9 (since i² = -1) = -66/9 - 34/9i
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Simplify: The final simplified form of (-2 - 1/3i)³ is -22/3 - 34/9i.
Key Points
- Complex Number Arithmetic: Remember to distribute and multiply carefully when working with complex numbers.
- Imaginary Unit: The imaginary unit 'i' is crucial to understanding and manipulating complex numbers.
- Simplification: Always aim to express complex numbers in the form a + bi, where a and b are real numbers.
This process illustrates the steps involved in simplifying complex numbers raised to powers. By understanding the properties of complex numbers and applying the rules of arithmetic, you can efficiently simplify such expressions.