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

Question

In Exercises 37–52, perform the indicated operations and write the result in standard form.

( −6 − √(−12)) / 48

Accepted Answer

Welcome back everyone. In this problem, we want to simplify the expression negative two plus the square root of negative 20 all divided by 32 and express our answer in standard form. If necessary for our answer choices, A is negative 1/16 plus the square of five divided by 16 I B is negative 5/8 plus the square of two divided by 32 IC is negative 1/16 minus the square of five divided by 16 I and D is negative 5/8 minus the square of two divided by 32. I. No, it says here we want to express our answer in standard form and that's the standard form for complex numbers. Recall that for complex numbers, the standard form is when we write our complex number as A plus B I where A is the real part and B is the imaginary part. And the reason it tells us that is because we have the square root of a negative number here. And by definition, that's what makes it a an imaginary number because the unit of an imaginary number I by definition equals the square of negative one. So if we can simplify and get that unit from our expression, then we should be able to write it in standard form. Now, before we continue notice that the square root of negative 20 can be simplified further. It's not in its simplest form as a radical. So let's think what are the square factors of 20? Well, four comes to mind. So I can rewrite this as negative two plus the square root of four multiplied by negative five. And that's all divided by 32. Now, how does that help us? Well, again, we can recall the product to, so we can also note that when we have the square root of our product A B, then it's the same thing as multiplying the square root of A by the square root of B. So we can apply that idea here because no, that tells us we can rewrite this as negative two plus the square root of four multiplied by the square root of negative five, all divided by 32. The square of four is two. So it's negative two plus two multiplied by the square root of negative five, all divided by 32. Know that the radical is in its simplest form, I can apply the same idea to rewrite it as A or rewrite it as an imaginary number, right? Because I can rewrite the square root of negative five as the square of five multiplied by negative one in doing that. Then my expression now becomes negative two plus two, multiplied by the square of five multiplied by the square of negative one. But again, by definition, the square of negative one is I, so I can replace that. Let me continue writing. And then in the next line, we can replace that with I. No, our number is almost in standard form. We just need to go ahead and divide through by 32. No negative two divided by 32 would be equal to negative 1/16 and two multiplied by the square of five I divided by 32. Again, two is a common factor. So that would be 1/16 of the square root of five I, which would just be written as a square of five divided by 16. I. So our final answer is negative 1/16 plus the square of five divided by 60 I. And we're sure it's in standard form because it has a real part, negative 1/16 and an imaginary part. The spirit of five divided by 16. When we look back in our answer choices that would have been answer choice. A 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.
( −6 − √(−12)) / 48

Key Concepts

Complex Numbers and Standard Form

Simplifying Square Roots of Negative Numbers

Operations with Complex Numbers

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