Question

What is the square root of $\left( { - 26} \right)$ times the square root of $\left( { - 13} \right)$?

Answer 1

Hint : In the given problem, we need to evaluate the multiplication of two square roots of negative numbers. The given question requires knowledge of the concepts of complex numbers and square roots. 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 solution:
In the question, we need to evaluate the product of the square root of $\left( { - 26} \right)$ and the square root of $\left( { - 13} \right)$ . For evaluating the product, we have to first simplify the expression and represent it in the form of complex numbers. We need to have an idea of complex numbers, their properties and how to do basic operations on the complex numbers set.
So, we have, $\sqrt {\left( { - 26} \right)} \times \sqrt {\left( { - 13} \right)}$
So, representing both the terms in the form of complex numbers, we get,
$\Rightarrow i\sqrt {26} \times i\sqrt {13}$
Now, simplifying the product, we get,
$\Rightarrow {i^2}\left( {\sqrt {26} \times \sqrt {13} } \right)$
Now, we know that the value of ${i^2}$ is $\left( { - 1} \right)$ . So, we get,
$\Rightarrow \left( { - 1} \right)\left( {\sqrt {26} \times \sqrt {13} } \right)$
Now, representing $26$ as product of its prime factors, we get,
$\Rightarrow \left( { - 1} \right)\left( {\sqrt {13} \times \sqrt {13} \times \sqrt 2 } \right)$
Now, we know that the value of $\sqrt {13} \times \sqrt {13}$ is $13$ . So, we get,
$\Rightarrow - 13\sqrt 2$
So, the value of the square root of $\left( { - 26} \right)$ times the square root of $\left( { - 13} \right)$ is $\left( { - 13\sqrt 2 } \right)$ .
So, the correct answer is “ $\left( { - 13\sqrt 2 } \right)$ ”.

Note : The given question involves solving the square root of a negative number and that’s where the set of complex numbers comes into picture and plays a crucial role in mathematics. The answer can be verified by working the solution backwards.