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.
           __
(−3 − √−7)²

Accepted Answer

Welcome back everyone. In this problem, we want to simplify the expression negative four minus the square root of negative 11 R squared. And we want to express our answer in standard form. If necessary for our answer choices, A is negative 40 plus 8, 8 IB is negative 40 minus + IC is five minus eight multiplied by the square of 11 I and the D is five plus eight multiplied by the square of 11. I. Now we want to simplify and write our answer in standard form. And that's a standard form for complex numbers. Recall that for a complex number, its standard form is when we write it in the form A plus B I where A is the real part and B is the imaginary part. And we're going to have to do that because we're finding the square root of a negative value here. So that is going to make it an imaginary number because as well recall that by definition, the unit for an imaginary number I is really the square root of negative one. So if we can go ahead and square our term here, then we will be able to write it out, we'll be able to expand it, sorry. And then we'll see if we can simplify it. Now, look at our expression here, doesn't it look a little familiar to you? Doesn't it look like some algebra that we've done on the perfect square? Again, recall that when we are squaring the difference of two terms, then the perfect square tells us that it will be equal to X squared minus two xy plus Y squared. In other words, we square both terms and then multiply both terms by two and subtract. So if we do that here, that means we can rewrite this as negative four R squared minus two, multiplied by negative four, multiplied by the negative square root of negative 11 or rather the square root of negative and added to the square root of negative 11 R squared. Now, when we do that negative four squared is 16, negative two, multiplied by negative four is a positive eight. So we'll have plus eight multiplied by the square of negative 11 and the square root of negative 11 squared where the square cancels the square root. So that means it would be equal to negative 11. We can combine like terms 16 and negative 11 to be five. So you're going to have five plus eight multiplied by the square of negative 11. No, we're not in standard form yet because we need to see the square root of negative one, the unit of an imaginary number clearly. And we can do that by rewriting negative 11 as 11 multiplied by negative one. By doing that, we can use the product property for radicals. And that property tells us that if we're finding the radical of our product A B is the same thing as multiplying the radical of A by the radical of B. So using that idea, we can rewrite this as five plus eight multiplied by the square of 11 multiplied by the square of negative one. Now we have our imaginary unit, we can take it a step further to write it as five plus eight multiplied by the spirit of 11. I no, we are in the standard form where five is the real part and eight multiplied by the script of 11 is the imaginary part. And when we look on our answer choices that would have been answer choice. D 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. __ (−3 − √−7)²

Key Concepts

Complex Numbers

Operations with Complex Numbers

Standard Form of Complex Numbers

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