(-3-4i)(7-5i).jpeg "(3i)(-3-4i)(7-5i)")
Multiplying Complex Numbers: (3i)(-3-4i)(7-5i)
This article explores the multiplication of complex numbers, specifically focusing on the expression (3i)(-3-4i)(7-5i).
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 (i² = -1).
The Multiplication Process
To multiply complex numbers, we follow the distributive property, similar to multiplying binomials in algebra:
-
Expand the product: (3i)(-3-4i)(7-5i) = (3i)(-21 + 15i - 28i + 20i²)
-
Simplify using i² = -1: = (3i)(-21 - 13i - 20) = (3i)(-41 - 13i)
-
Distribute again: = -123i - 39i²
-
Substitute i² = -1 and simplify: = -123i + 39
-
Express in standard form (a + bi): = 39 - 123i
Conclusion
Therefore, the product of (3i)(-3-4i)(7-5i) is 39 - 123i. This process highlights the importance of understanding the properties of complex numbers, particularly the value of i², to simplify expressions and arrive at the final complex number form.