Question

Find out the product of the complex number \[\left( {8 - 2i} \right)\] and its conjugate.

Answer 1

Hint: The given question deals with the concept of multiplying the given complex number with its conjugate. To solve the question we will at first find out the conjugate of the complex number and multiply it with its conjugate using the distributive law.

Complete step-by-step solution:
Before dwelling on the question, let us first understand what are complex numbers and what are conjugates.
Let us denote \[\sqrt { - 1} \] by \[i\]. Then we get, \[{i^2} = - 1\].
The above observation clearly means \[i\] that is the solution of the equation \[{x^2} + 1 = 0\].
 Any number of forms \[p + qi\], where p and q are real numbers, is defined as a complex number. For example: \[3 + 2i\], \[5 + i\sqrt 6 \]are both complex numbers.
For any complex number \[z = p + qi\], \[p\] is the real part denoted by Re \[z\]and \[q\] is called the imaginery part denoted by Im \[z\].
Now, let \[z = p + qi\] be a complex number. Then its conjugate, denoted by \[\overline z \], is the complex number \[\overline z = p - qi\]. Simply put, the conjugate of any complex number can be found out just by altering the sign of the imaginary part.
Let us now proceed with the given question,
Given complex number is, \[\left( {8 - 2i} \right)\]
Therefore, the conjugate of the given complex number is \[\overline z = 8 + 2i\].
Now, in the given problem we are asked to find the product of the given complex number and its conjugate.
Therefore, we have,
 \[ \Rightarrow \left( {8 - 2i} \right)\left( {8 + 2i} \right)\]
Applying distributive law of multiplication, we get,
\[ \Rightarrow 8 \times 8 + 8 \times 2i - 8 \times 2i - 2i \times 2i\]
\[ \Rightarrow 8 \times 8 + 8 \times 2i - 8 \times 2i - 4{i^2}\]
\[ \Rightarrow 64 + 16i - 16i - 4{i^2}\]
\[ \Rightarrow 64 - 4{i^2} - - - - - \left( 1 \right)\]
Now, we know that \[i = \sqrt { - 1} \]
Thus, \[{i^2} = - 1\]
Therefore, putting the value \[i\] of in equation (1)
We get, \[ \Rightarrow 64 - 4( - 1)\] which implies,
\[ \Rightarrow 64 + 4 = 68\] which is our required product.

Thus the correct answer is \[ 64 + 4 = 68\].

Note: The distributive law in mathematics is an operation of multiplication and addition. Under the distributive law, each term of one bracket or parentheses is multiplied with every other term of every other of the parentheses. For example: \[k(p + q) = kp + kq\]i.e., the monomial factor k is multiplied or applied to every other factor of the binomial \[(p + q)\].