Question

What is the square root of $-50$ times the square of $-10$ ?

Answer 1

Hint: To calculate the product of two square roots, we can either calculate the two square roots separately or we can calculate the product of the two numbers and then find out the square root. We shall proceed in our solution by applying the first method. This will give us the required solution to our problem.

Complete step by step answer:
Let us first assign some terms to the quantities given in our problem. Let us say the square root of -50 is denoted by ‘x’ and the square root of -10 is denoted by ‘y’ and the product of their combined square roots is represented by ‘z’. Then, we need to find the value of ‘z’.

According to our above definition of terms, we can write:
$\Rightarrow x=\sqrt{-50}$
$\Rightarrow x=5\sqrt{2}i$
Let us name the above equation as (1), so we have:
$\Rightarrow x=5\sqrt{2}i$ .......... (1)

Now, for our next term, we can write:
$\Rightarrow y=\sqrt{-10}$
$\Rightarrow y=\sqrt{10}i$
Let us name the above equation as (2), so we have:
$\Rightarrow y=\sqrt{10}i$ .......... (2)

And lastly, we can write the product of these two terms as:
$\Rightarrow z=x\times y$
Let us name the above equation as (3), so we have:
$\Rightarrow z=x\times y$ .......... (3)

Now, putting the values of ‘x’ and ‘y’ from equation number (1) and (2) respectively and calculating for ‘z’, we get:
$\begin{align}
  & \Rightarrow z=5\sqrt{2}i\times \sqrt{10}i \\
 & \Rightarrow z=5\sqrt{20}{{\left( i \right)}^{2}} \\
 & \Rightarrow z=10\sqrt{5}\left( -1 \right) \\
 & \therefore z=-10\sqrt{5} \\
\end{align}$

Hence, the square root of -50 times the square root of -10 comes out to be $-10\sqrt{5}$ .

Note: While calculating the square roots of two numbers, we should always check whether the numbers are positive or negative. As while calculating the square root of a negative number, the result is always followed by a term called ‘iota’ which is equal to $\sqrt{-1}$ .