Question

How do you multiply \[ - 11i\left( {3 + 9i} \right)\] ?

Answer 1

Hint: Complex numbers, as any other numbers, are added, subtracted, multiplied or divided, and then those expressions can be simplified. And here to multiply the given complex number; we need to distribute the terms to remove the parentheses, then simplify the powers of i and next combine like terms i.e., combine real numbers with real numbers and imaginary numbers with imaginary numbers.

Complete step by step solution:
Given expression:
 \[ - 11i\left( {3 + 9i} \right)\]
We need to distribute the terms of the given expression to remove the parentheses:
 \[ \Rightarrow - 33i - 99{i^2}\]
Simplify the powers of i, specifically we must note that \[{i^2} = - 1\] , hence we get:
 \[ \Rightarrow - 33i - 99\left( { - 1} \right)\]
Combine like terms, i.e., combine real numbers with real numbers and imaginary numbers with imaginary numbers as:
 \[ \Rightarrow 99 - 33i\]
Therefore, we get
 \[ - 11i\left( {3 + 9i} \right) = 99 - 33i\]
So, the correct answer is “99 - 33i”.

Note: A complex number is a number that can be expressed in the form \[a + bi\] , where a and b are real numbers and i is the imaginary unit, that satisfies the equation \[{i^2} = - 1\] . In this expression, a is the real part and b is the imaginary part of the complex number.
We must note that to multiply a complex number by a real number we need to just multiply both parts of the complex number by the real number. Although real numbers are subsets of complex numbers and hence the sum of two complex numbers is always a complex number. To multiply monomials, multiply the coefficients and then multiply the imaginary numbers i and to multiply complex numbers that are binomials, use the Distributive Property of Multiplication.