Question

How do you simplify \[\left( {5 - 7i} \right)\left( { - 4 - 3i} \right)\]?

Answer 1

Hint: In solving the question, first us the distributive property, then simplify the powers of \[i\], specifically remember that \[{i^2} = - 1\], then combine like terms that is combine real numbers with real numbers and imaginary numbers with imaginary numbers.

Complete step-by-step solution:
Complex numbers are made of two types of numbers i.e., real numbers and imaginary numbers.
Complex numbers are defined by their inclusion of the \[i\] term, which is the square root of minus one. In basic-level mathematics, square roots of negative numbers don’t really exist, but they occasionally show up in algebra problems. The general form for a complex number is,
\[z = a + bi\], where \[z\] is the complex number, \[a\] represents any number, and \[b\] represents another number called the imaginary part, both of which can be positive or negative.
Given expression is \[\left( {5 - 7i} \right)\left( { - 4 - 3i} \right)\],
Now using FOIL method to perform multiplication, we get,
Steps of foil method will be: First multiply the first terms, then the outer terms, then the inner terms and finally the last terms.
Now multiplying the two terms using FOIL method we get,
$\Rightarrow$\[\left( {5 - 7i} \right)\left( { - 4 - 3i} \right) = 5 \cdot - 4 + 5 \cdot \left( { - 3i} \right) + \left( { - 7i} \right) \cdot \left( { - 4} \right) + \left( { - 7i} \right) \cdot \left( { - 3i} \right)\],
Now multiplying each term we get,
$\Rightarrow$\[\left( {5 - 7i} \right)\left( { - 4 - 3i} \right) = - 20 - 15i + 28i + 21{i^2}\],
Now combining the like terms we get,
$\Rightarrow$\[\left( {5 - 7i} \right)\left( { - 4 - 3i} \right) = - 20 + 13i + 21{i^2}\],
Now we know that\[{i^2} = - 1\],
$\Rightarrow$\[\left( {5 - 7i} \right)\left( { - 4 - 3i} \right) = - 20 + 13i + 21\left( { - 1} \right)\],
Now by simplifying we get,
$\Rightarrow$\[\left( {5 - 7i} \right)\left( { - 4 - 3i} \right) = - 20 + 13i - 21\],
Now adding the combined terms we get,
$\Rightarrow$\[\left( {5 - 7i} \right)\left( { - 4 - 3i} \right) = - 41 + 13i\].
The simplified form of \[\left( {5 - 7i} \right)\left( { - 4 - 3i} \right)\] is \[ - 41 + 13i\].

\[\therefore \]The simplified term of the given complex number will be \[ - 41 + 13i\].

Note: Complex numbers are a combination of real and imaginary numbers, and when two complex numbers multiply, the first complex number gets multiplied by each part of the second complex number.