(-3-7i).jpeg "(5+5i)(-3-7i)")
Multiplying Complex Numbers: (5 + 5i)(-3 - 7i)
This article will walk you through the process of multiplying two complex numbers: (5 + 5i) and (-3 - 7i).
Understanding Complex Numbers
Complex numbers are numbers that can be expressed in the form a + bi, where:
- a and b are real numbers.
- i is the imaginary unit, defined as the square root of -1 (i² = -1).
Multiplying Complex Numbers
To multiply complex numbers, we use the distributive property (also known as FOIL method), just like with binomials:
(a + bi)(c + di) = ac + adi + bci + bdi²
Since i² = -1, we can simplify the expression:
(a + bi)(c + di) = (ac - bd) + (ad + bc)i
Applying the Formula
Let's apply this to our problem: (5 + 5i)(-3 - 7i)
-
Identify a, b, c, and d:
- a = 5
- b = 5
- c = -3
- d = -7
-
Substitute the values into the formula:
- (5)(-3) + (5)(-7)i + (5)(-7)i + (5)(-7)i²
-
Simplify:
- -15 - 35i - 35i - 35(-1)
-
Combine real and imaginary terms:
- (-15 + 35) + (-35 - 35)i
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Final answer:
- 20 - 70i
Conclusion
Therefore, the product of (5 + 5i) and (-3 - 7i) is 20 - 70i. Remember that complex numbers are not just abstract mathematical concepts but have applications in fields like electrical engineering, physics, and signal processing.