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