How do you simplify $ {\left( { - 3 - 4i} \right)^2} $ ?

Hint:In this question, we need to simplify $ {\left( { - 3 - 4i} \right)^2} $ . Here, we will rewrite the term $ {\left( { - 3 - 4i} \right)^2} $ . Then we will use the FOIL method then also distributive property to evaluate it. Then we will substitute the value of $ {i^2} $ by which we will get the required solution.

Complete step-by-step solution:
Now, we can write $ {\left( { - 3 - 4i} \right)^2} $ as $ \left( { - 3 - 4i} \right)\left( { - 3 - 4i} \right) $
Let us expand $ \left( { - 3 - 4i} \right)\left( { - 3 - 4i} \right) $ using FOIL method,
= $ - 3\left( { - 3 - 4i} \right) - 4i\left( { - 3 - 4i} \right) $

To ‘distribute’ means to divide something or give a share or part of something. According to the distributive property, multiplying the sum of two or more addends by a number will give the same result as multiplying each addend individually by the number and then adding the products together
i.e., $ a\left( {b + c} \right) = ab + ac $ .

Therefore, by using the distributive property,
 $ = \left( { - 3 \times - 3} \right) + \left( { - 3 \times - 4i} \right) + \left( { - 4i \times - 3} \right) + \left(
{ - 4i \times - 4i} \right) $
 $ = 9 + 12i + 12i + 16{i^2} $
 $ = 9 + 24i + 16{i^2} $

An imaginary number is a complex number that can be written as a real number multiplied by the imaginary unit $ i $ , which is defined by its property $ {i^2} = - 1 $ .

Therefore, by applying the value of $ {i^2} = - 1 $ , we have,
 $ = 9 + 24i + 16 \times - 1 $
 $ = 9 + 24i - 16 $
 $ = - 7 + 24i $

Therefore, by simplifying $ {\left( { - 3 - 4i} \right)^2} $ we get $ - 7 + 24i $ .

Note: In this question, it is important to note here that the FOIL method is a technique for distributing two binomials. The letters FOIL stands for First, Outer, Inner, Last. First means multiply the terms which occur first in each binomial, Outer means multiply the outermost terms in the product, Inner means
multiply the innermost terms, and Last means multiply the terms which occur last in each binomial. The distributive property helps in making difficult problems simpler. The distributive property of multiplication can be used to rewrite expression by distributing or breaking down a factor as a sum or difference of two numbers.