Question

How do you simplify $ {( - 1 + 6i)^2} $ ?

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 $ ( - 1 + 6i) $ . So, in order to evaluate the answer to the given question, we use the algebraic identity to find the square of the binomial $ {\left( {a + b} \right)^2} = \left( {{a^2} + 2ab + {b^2}} \right) $ and $ {\left( {a - b} \right)^2} = \left( {{a^2} - 2ab + {b^2}} \right) $ , we get,
So, $ {( - 1 + 6i)^2} = {(6i - 1)^2} $
 $ = {(6i)^2} - 2(6i)(1) + {(1)^2} $
 $ = 36{i^2} - 12i + 1 $
Now, we know that $ {i^2} = - 1 $ . So, substituting the value of $ {i^2} $ , we get,
 $ = 36\left( { - 1} \right) - 12i + 1 $
Further simplifying the calculation, we get,
 $ = - 12i - 35 $
So, we get the value of $ {( - 1 + 6i)^2} $ as $ \left( { - 12i - 35} \right) $ .
So, the correct answer is “ $ - 12i - 35 $ ”.

Note: The given question revolves around solving the square of a complex number 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. The answer can be verified by working the solution backwards and observing that square root of $ \left( { - 12i - 35} \right) $ is $ ( - 1 + 6i) $ . Algebraic identities $ {\left( {a + b} \right)^2} = \left( {{a^2} + 2ab + {b^2}} \right) $ and $ {\left( {a - b} \right)^2} = \left( {{a^2} - 2ab + {b^2}} \right) $ help us to compute the squares of binomial expressions.