%5E5.jpeg "(2i)^5")
Understanding (2i)^5
This article aims to explain how to simplify the expression (2i)^5, focusing on the properties of imaginary numbers and exponents.
Imaginary Numbers and Exponents
i represents the imaginary unit, defined as the square root of -1. It's crucial to remember that i² = -1.
Exponents indicate repeated multiplication. For example, (2i)^5 means multiplying (2i) by itself five times:
(2i)^5 = (2i) * (2i) * (2i) * (2i) * (2i)
Simplifying the Expression
Let's break down the simplification:
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Separate the real and imaginary components: (2i)^5 = (2^5) * (i^5)
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Calculate the real component: 2^5 = 32
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Simplify the imaginary component: i^5 = i^4 * i Since i^4 = (i^2)^2 = (-1)^2 = 1, we get: i^5 = 1 * i = i
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Combine the components: (2i)^5 = 32 * i = 32i
Conclusion
Therefore, (2i)^5 simplifies to 32i. This process demonstrates how to handle exponents with imaginary numbers, utilizing the key property of i² = -1.