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Multiplying Complex Numbers: (3 + 5i)(5 - 3i)
This article explores the multiplication of complex numbers, specifically focusing on the expression (3 + 5i)(5 - 3i).
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.
Multiplication of Complex Numbers
Multiplying complex numbers is similar to multiplying binomials. We use the distributive property (also known as FOIL method) to expand the expression:
(3 + 5i)(5 - 3i) = (3 * 5) + (3 * -3i) + (5i * 5) + (5i * -3i)
Simplifying each term:
= 15 - 9i + 25i - 15i²
Remember that i² = -1. Substituting this value:
= 15 - 9i + 25i + 15
Combining real and imaginary terms:
= (15 + 15) + (-9 + 25)i
= 30 + 16i
Final Result
Therefore, the product of (3 + 5i)(5 - 3i) is 30 + 16i.
Key Points
- Complex numbers are expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit.
- Multiplication of complex numbers follows the distributive property.
- Remember that i² = -1.
This example demonstrates how to multiply complex numbers. By applying the distributive property and simplifying, we arrive at the final result, which is also a complex number.