Question

How do you simplify $\left( {1 + 5i} \right)\left( {1 - 5i} \right)$ ?

Answer 1

Hint : In the given problem, we need to evaluate the square of a given complex number. The given question requires knowledge of the concepts of complex numbers and how to perform operations like squaring the complex number. The square root of a negative number is always a complex number. Hence, we must have in mind the definition of complex numbers and their basic properties.

Complete step-by-step answer :
The given problem requires us to find the square of the given complex number $(2 - 3i)$ . So, in order to evaluate the answer to the given question, we use the algebraic identity to find the difference of squares of two terms ${a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)$ , we get,
So, $\left( {1 + 5i} \right)\left( {1 - 5i} \right)$
$= {(1)^2} - {(5i)^2}$
$= 1 - 25{i^2}$
Now, we know that ${i^2} = - 1$ . So, substituting the value of ${i^2}$ , we get,
$= 1 - 25\left( { - 1} \right)$
Further simplifying the calculation, we get,
$= 1 + 25$
$= 26$
So, we get the value of $\left( {1 + 5i} \right)\left( {1 - 5i} \right)$ as $26$ .
So, the correct answer is “26”.

Note : The given question revolves around simplifying the product of two terms both involving complex numbers and that’s where the set of complex numbers comes into picture and plays a crucial role in mathematics. Algebraic rules and operations are also of great significance and value when it comes to simplification of expressions. We must have a good grip on algebraic simplification along with knowledge of properties of complex numbers and identities so as to tackle this kind of problems with ease