Video Transcript
Simplify the square root of negative 61.
In this question, we’re asked to simplify the square root of a number. However, we can see that this number is negative. So to do this, we’re first going to need to recall how we take the square root of a negative number. And we do this by using our imaginary number 𝑖, which is a solution to the equation 𝑥 squared is equal to negative one. In other words, we can think of 𝑖 as being the square root of negative one.
So to use this to simplify our expression, we’re first going to need to rewrite our expression in terms of the square root of negative one. We’ll start by rewriting negative 61 as negative one multiplied by 61. Then, all we need to do is remember the square root of 𝑎 times 𝑏 is equal to the square root of 𝑎 multiplied by the square root of 𝑏. So we can take the square root of each of these factors separately. This gives us the square root of negative one multiplied by the square root of 61. And we already know we can rewrite the square root of negative one as 𝑖.
However, we should also check whether we can simplify the square root of 61. Remember, to do this, we need to see if 61 has any square factors. However, if we were to look at the factors of 61, we would find out that 61 is a prime number. So it has no square factors. So we’ll just leave this as the square root of 61.
So we could leave our answer like this. However, we can see this is an imaginary number. And usually, we write the coefficient of 𝑖 at the front. So we will reorder these two factors to get our final answer of the square root of 61 multiplied by 𝑖. And we could stop here. However, there is one thing worth pointing out about this example. Remember, when we take the square root of a positive number, we always take the positive square root. So the square root of four is equal to two. It’s not equal to negative two.
Something very similar is true when we’re taking the square root of a negative number. We say that the square root of negative one is equal to 𝑖. But in fact, there’s two numbers which square to give us negative one. Negative 𝑖 all squared is also equal to negative one. This in fact leaves us with the exact same problem we had when we’re taking the square root of a positive number. We can find two solutions to the equation 𝑥 squared is equal to negative 61. Either 𝑥 is equal to the square root of 61 times 𝑖 or 𝑥 is equal to negative the square root of 61 multiplied by 𝑖. And we get around this in exactly the same way we did with the real case. We just define the square root of negative one to be equal to 𝑖. This is why the answer to this question is the square root of 61𝑖 and is not negative one times the square root of 61𝑖.
Therefore, we were able to simplify the square root of negative 61. We got that it’s equal to the square root of 61 times 𝑖.