%5E2%2B5(3-5i).jpeg "(4-i)^2+5(3-5i)")
Simplifying Complex Expressions: (4-i)² + 5(3-5i)
This article will guide you through the steps of simplifying the complex expression (4-i)² + 5(3-5i).
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: (4-i)² = (4-i)(4-i) = 16 - 4i - 4i + i² = 16 - 8i - 1 = 15 - 8i
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Distribute the 5: 5(3-5i) = 15 - 25i
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Combine like terms: (15 - 8i) + (15 - 25i) = 30 - 33i
Therefore, the simplified form of the expression (4-i)² + 5(3-5i) is 30 - 33i.
Conclusion
Simplifying complex expressions involves applying the basic rules of algebra and understanding the properties of imaginary numbers. By following the steps outlined above, you can effectively simplify expressions involving complex numbers.