Operations with Square Roots with Negative Numbers in The Radicand

Hi. Here I'm going to show you an example of how to simplify an expression when you have a negative inside of a square root. In the example we have here, we have 5 minus the square root of negative 49, that whole amount squared. And this negative is what we want to deal with first. For real numbers, that just isn't going to work for us. So in order to do the rest of these calculations, to continue simplifying, the first thing we're going to want to do is go ahead and convert this into a complex number, meaning we want to take that amount and write it in terms of i. It makes all of the calculations much simpler. So first off, recognize the fact that the square root of negative 49 is the same as the square root of 49 times the square root of negative 1. And that's just thanks to the properties for radicals, including the square roots we have here. Both of these can simplify. The first one, the square root of 49, well, that's 7 because 49 is a perfect square. The square root of negative 1-- in the real numbers, we can't do anything with that. But in complex numbers, with imaginary numbers, we can, by definition, replace that with the number i. So instead of having this be the square root of negative 49, we're going to rewrite it as 7i. Again, makes everything much easier to deal with. So the problem we want to look at now is 5 minus 7i, that amount squared. Now it may be very tempting to look at this and see we have an amount squared and just square both of them, say that this is 5 squared and 7i squared. But that doesn't work. In order to simplify this, we're actually going to need to FOIL this out. So instead of writing it like this, it may be easier to write it in an expanded form, meaning write this as 5 minus 7i times 5 minus 7i. If you write both of them out like this, it's easier to see that we're going to need to FOIL this in order to simplify. So start multiplying. 5 times 5 is going to give us 25. 5 times negative 7i is going to give us minus 35i. The next one, we have another negative 7i times 5, so minus 35i again. And the last two, negative 7i times negative 7i will be plus 49i squared. Now we're going to treat this pretty much the same way we would if these have been variables instead of i's and go ahead and combine any like terms that we happen to see. So looking at it here, you notice we have negative 35i, negative 35i. Those are like terms, so we can go ahead and combine those. So we end up with 25 minus 70i plus 49i squared. Next up, to continue this process, we recognize the fact that you don't ever, ever, ever want to leave a final answer with i squared in it. And just in case you didn't quite catch that, I'll say it one more time. Don't ever leave i squared in your final answer. We're going to use the definition of i to take this part out, this I squared out, and see that it's really the number negative 1. So we'll take it out, put negative 1 in instead, and simplify from there. That gives us 25 minus 70i plus 49 times negative 1, not i squared. Well, the 25 here and the 49 times negative 1, that's negative 49. Those are both constants. They are like terms, and we can go ahead and combine those two together, which is going to give us a final answer of negative 24 minus 70i. Remember, when dealing with complex numbers, you always want to leave your final answer in standard form, which means in the form of a plus bi, and that's what we have right here. So our final answer, negative 24 minus 70i.