Question

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

Answer 1

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.