In Exercises 29–36, simplify and write the result in standard for...

Question

In Exercises 29–36, simplify and write the result in standard form.  ____√−108

Accepted Answer

Welcome back. Everyone. In this problem, we want to simplify the expression of the square root of negative 245 expressing our answer in standard form. For our answer choices A is five multiplied by the square root of seven I B is negative five multiplied by the square of seven IC is seven multiplied by the square of five I and D is negative seven multiplied by the square of five I. No. It says here we want to write our answer in standard form. And that's a standard form for complex numbers. And recall that for complex numbers, writing it in standard form means that we're writing it in the form A plus B I where A is a real number and B is an imaginary number. And the reason we need to do that is because we're finding the square root of a negative number. So by definition also recall that the unit for an imaginary number I equals the square root of negative one. So if we can factor out negative one from our expression, then we will be able to show that it is an imaginary number. No, we're dealing with 245. And before we try to factor out negative one, it would be a good idea to write or radical in the simplest form possible that is without any square factors. So let's think about it. Are there any square numbers that we can multiply by to get 245? Well, we can multiply by 49. So we could rewrite the square root of negative 245 as the square root of 49 multiplied by five multiplied by negative one. No. At this point, we can also get some help with the product rule for radicals because again, recall that for the radical of a product A B is the same thing as the radical of A multiplied by the radical of B which it makes sense that even if we have three terms, the same idea can still apply. In other words, we can rewrite our expression as the square root of multiplied by the square root of five multiplied by the square root of negative one. Now, by definition, we've shown that the square root of negative one is a factor. So we can replace it with I, we can also find the square root of 49 which is seven. So our answer will be seven multiplied by the square of 55 has no square factors. So it's in its simplest form as a radical and all of that will be multiplied by I no, our expression is in standard form because even though there's no real part, there's an imaginary part. So our answer is seven multiplied by the square root of five. 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 29–36, simplify and write the result in standard form. ____√−108

Key Concepts

Complex Numbers

Square Roots of Negative Numbers

Standard Form of Complex Numbers

Watch next