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. I'm so glad you're here. We've got this expression, I'll rewrite it here. It's negative five plus the square root of negative 25 all squared and we are to simplify that and then write the answer in standard form. What is the standard form here? We are dealing with the square root of a negative number, meaning it's a complex number. And the standard form of a complex number is A plus B. I. Or rewriting this all. In terms of I. And we recall that I equals the square root of negative one. So now let's work on getting that squared of negative 25 rewritten in terms of I. So I'll rewrite this as negative five plus the square root of negative one times squared. And now we can see that square root of negative one in there and we can bring that out as an eye. So we've got negative five plus I times the square root of 25 all squared. And now, what else can we do to simplify? Well, we know that the square root of 25. We could break that down into a factor tree because 25 is five times five or five squared. We can rewrite that here as negative five plus I times the square root of five squared. Which is exciting if you're five squared because the square root of five squared is five. So this is negative five plus five. I all squared. All right. Now we can get to work on this, we can rewrite this as negative five plus five, I times negative five plus five. I now we can do our good old fashioned foiling negative five times negative five is positive 25 those are our first. Now our Outsides negative five times positive five I. Is negative 25. I. And we do our in our insides F. O. I positive five times negative five is negative 25. I again and our last C. O. I. L. Positive five times positive five. I. Is plus 25 I squared. Alright, what can we do? Well, we've got some like terms here negative 25 I minus 25. I let's combine those. Bring down our 25 negative 25 I minus 25 I is minus 50 I. And our plus 25 I squared. When we look at that I squared and we recall that by definition I squared is negative one. So that's going to be plus 25 times negative one. Go ahead and do that. Multiplication. 25 times negative one. 25 -50 i 25 times -1 is -25. Now we can combine these leg terms 25 minus 25. Well 25 minus 25. That just cancels out that zero. So all we're left with is negative 50 I. And we look at our answer choices and that matches with answer choice B Well done. We'll catch on the next one

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

Key Concepts

Complex Numbers

Standard Form of Complex Numbers

Binomial Expansion

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