(-2%2B4i).jpeg "(3-5i)(-2+4i)")
Multiplying Complex Numbers: (3-5i)(-2+4i)
This article will guide you through the process of multiplying two complex numbers: (3-5i) and (-2+4i).
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.
Multiplying Complex Numbers
To multiply complex numbers, we use the distributive property, just like we do with real numbers. This means we multiply each term in the first complex number by each term in the second complex number.
Step-by-Step Solution
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Expand using the distributive property:
(3-5i)(-2+4i) = 3(-2) + 3(4i) - 5i(-2) - 5i(4i)
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Simplify by multiplying:
= -6 + 12i + 10i - 20i²
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Remember that i² = -1:
= -6 + 12i + 10i - 20(-1)
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Combine real and imaginary terms:
= (-6 + 20) + (12 + 10)i
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Final Result:
= 14 + 22i
Therefore, the product of (3-5i) and (-2+4i) is 14 + 22i.
Conclusion
Multiplying complex numbers is a straightforward process that involves applying the distributive property and remembering the fundamental property of the imaginary unit (i² = -1).