In Exercises 37–52, perform the indicated operations and write th...

Question

In Exercises 37–52, perform the indicated operations and write the result in standard form.
           __
(−2 + √−4)²

Accepted Answer

Welcome back everyone. In this problem, we want to simplify the expression negative five plus the square root of negative 25 all squared. And we want to express our answer in standard form. If necessary for our answer choices, A is negative 25 IB is 25. IC is negative 50 I and D is 50. I no. We want to express our answer in standard form. And what that really means is that we want to write our result for a complex number in the form A plus B I where A is the real part and B is the imaginary part. And we'll need to do that because we're finding the square root of a negative number in our expression. And by definition, that's what a complex number is the unit of a complex number I OK equals the square root of negative one. So if we can show that explicitly in our problem in our expression, then we'll be able to write it in standard form. So let's go ahead and do that. So we're squaring negative five plus the square of negative 25. But let's think about it a little. Doesn't this look like something you've seen before. Doesn't this look like a perfect square and a perfect square? The property for a perfect square like you would have done in algebra? Recall that it tells you that if you're finding the square of a sum of two expressions of a binomial, it means that we can rewrite it as the square of both terms and the two multiplied by both of those terms. So it would have been X squared plus two xy plus Y squared. So if we apply that same idea here, it means that we can rewrite this as negative five squared plus two, multiplied by those terms, two multiplied by negative five, multiplied by the square root of negative 25 plus the square of the last term, the square root of negative 25 or squared. So now that we have that know that we've applied our property, let's go ahead and simplify negative five squared equals 25 2, multiplied by negative five is negative 10 and negative multiplied by the square root of negative 25 will be our second term. And for a third term, the square root of negative 25 squared, the square cancels the square root. So that would just leave us with minus 25. Notice that we have two like terms 25 and negative 25 or we combine them 25 minus 25 equals zero. So we can rewrite this as negative 10 multiplied by the square root of negative 25 but it's still not in the standard form. How can we do that? Well, first, let's try to simplify our radical and that's because or radical term can be rewritten as or we can rewrite it. So a square factor, it's not in its simplest form because it has a square factor here. So let's do that. Let's rewrite the spirit of negative 25 as 25 multiplied by negative one. No, if you recall again, a property that we use with radicals, recall that the square root of our product A B equals the square of A multiplied by the square of B. Using that idea, we can rewrite what's under our square root as a square of multiplied by the spirit of negative one. No, the square of 25 is five. So we'll have negative 10 multiplied by five and the square root of negative one by definition is I. So we can rewrite an I in place of the square root of negative one. Finally negative 10 multiplied by five is negative 50. So our answer is negative 50. I it's in standard form because even though there's no real part, we have an imaginary part negative 50 I. And when we look in our answer choices that would have been answer choice. C thanks a lot for watching everyone. I hope this video helped

In Exercises 37–52, perform the indicated operations and write the result in standard form. __ (−2 + √−4)²

Key Concepts

Complex Numbers

Operations with Complex Numbers

Standard Form of Complex Numbers

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