%E2%8B%85(%E2%88%922%E2%88%925i).jpeg "(4+4i)⋅(−2−5i)")
Multiplying Complex Numbers: (4 + 4i) ⋅ (-2 - 5i)
This article will explore the multiplication of complex numbers, focusing on the specific example of (4 + 4i) ⋅ (-2 - 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.
Multiplying Complex Numbers
To multiply complex numbers, we can use the distributive property, similar to how we multiply binomials in algebra.
Step 1: Expand the product
(4 + 4i) ⋅ (-2 - 5i) = 4(-2 - 5i) + 4i(-2 - 5i)
Step 2: Distribute
= -8 - 20i - 8i - 20i²
Step 3: Simplify using i² = -1
= -8 - 28i - 20(-1)
Step 4: Combine real and imaginary terms
= -8 + 20 - 28i
Step 5: Final result
= 12 - 28i
Conclusion
Therefore, the product of (4 + 4i) ⋅ (-2 - 5i) is 12 - 28i. This demonstrates how to multiply complex numbers using the distributive property and the fundamental property of the imaginary unit, i² = -1.