(%E2%88%929%2B3i).jpeg "(−8+10i)(−9+3i)")
Multiplying Complex Numbers: (−8+10i)(−9+3i)
This article will guide you through the process of multiplying two complex numbers: (−8+10i) and (−9+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 (i² = -1).
Multiplication Process
To multiply two complex numbers, we use the distributive property (also known as FOIL method). This involves multiplying each term of the first complex number by each term of the second complex number.
Step 1: Expand the Expression
(−8+10i)(−9+3i) = (−8)(-9) + (−8)(3i) + (10i)(-9) + (10i)(3i)
Step 2: Simplify
- (−8)(-9) = 72
- (−8)(3i) = -24i
- (10i)(-9) = -90i
- (10i)(3i) = 30i²
Step 3: Substitute i² with -1
- 30i² = 30(-1) = -30
Step 4: Combine Real and Imaginary Terms
72 - 24i - 90i - 30 = (72 - 30) + (-24 - 90)i
Step 5: Final Result
(−8+10i)(−9+3i) = 42 - 114i
Conclusion
Therefore, the product of the complex numbers (−8+10i) and (−9+3i) is 42 - 114i. Remember, when multiplying complex numbers, you are essentially combining real and imaginary components, taking into account the property of the imaginary unit 'i'.