91Ó°ÊÓ

Multiplying complex numbers involves interacting both the real and imaginary components of each number. Complex numbers usually appear in the form \( a + bi \), where \( a \) is the real part, and \( bi \) is the imaginary part with \( i \) being \( \sqrt{-1} \). When multiplying complex numbers like \((7 - 5i)(-2 - 3i)\), each component must be combined using the distributive property. You treat the expression like multiplying two binomials in algebra. This means taking each term from the first complex number and multiplying it by each term from the second. The process results in four separate products that need to be calculated and summarized in the answer.

The standard form of a complex number is expressed as \( a + bi \). This makes it easier to understand the real and imaginary parts of the number.
After performing the multiplication of the complex numbers, the terms need to be rearranged to fit this format. This involves grouping the real terms together, and the imaginary terms together as well. For instance, if after multiplication you obtain \(-14 - 21i + 10i + 15\), you group the real numbers \(-14\) and \(15\), resulting in \(1\), and then combine the imaginary terms \(-21i + 10i\) to get \(-11i\). Thus, you arrive at the final standard form of \(1 - 11i\).
The standard form is essential for ensuring that complex numbers are consistently presented, which makes them easier to interpret and work with in further mathematical applications.

The distributive property is a key rule in algebra and is pivotal when multiplying complex numbers. It states that \( a(b + c) = ab + ac \). This property is used to multiply out the terms within each complex number, as was done in this example with \( (7-5i)(-2-3i) \).
By applying the distributive property here, you multiply each component of one complex number by each component of the other. This expands \(7(-2-3i)\) to \(-14 - 21i\), and \((-5i)(-2-3i)\) to \(10i + 15i^2\).
Remember, \(i^2 = -1\), so \(15i^2 = 15(-1) = -15\), modifying the real part during simplification. Using the distributive property ensures that every part of both complex numbers is accounted for in the multiplication process, preparing them for combination and simplification into the standard form.