Question

multiply -1 - 9i by its complex conjugate. the product is \square.

Explanation:

Step1: Find the complex conjugate

The complex conjugate of a complex number \(a + bi\) is \(a - bi\). For \(-1 - 9i\), its complex conjugate is \(-1 + 9i\).

Step2: Multiply the two complex numbers

We use the formula \((x + y)(x - y)=x^2 - y^2\) (which is a special case of the distributive property, also known as the difference of squares). Here, \(x=-1\) and \(y = 9i\).
First, expand \((-1 - 9i)(-1 + 9i)\) using the formula \((a + b)(a - b)=a^2 - b^2\), where \(a=-1\) and \(b = 9i\).
So, \((-1)^2-(9i)^2\).
Calculate \((-1)^2 = 1\) and \((9i)^2=9^2\times i^2 = 81\times(- 1)=-81\) (since \(i^2=-1\)).
Then, \(1-(-81)=1 + 81=82\).

Answer:

\(82\)