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.
    ___          ___
5√−16  +  3√−81

Accepted Answer

Welcome back everyone. In this problem, we want to simplify the expression four multiplied by the square root of negative 25 plus two, multiplied by the square root of negative 81. And we want to express our answer in standard form for our answer choices A is two. IB is 38. IC is 46 I and D is 26. I. Now, when it says that we want to express our answer in standard form, it means the standard form for complex numbers. And recall that for a complex number, writing it in standard form means that we're writing it in a 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 dealing with the square root of negative numbers. And by definition, we can also recall that I, the imaginary unit equals the square root of negative one. So if I can factor out the square root of negative one in these expressions, then I'll be able to rewrite the number in standard form. So let's do that. No negative 25 for four, negative 25 I can write it as 25 multiplied by negative one. Likewise, for negative 81 I can rewrite it as 81 multiplied by negative one. Now, we can use what we know about the product of the radical or the product of radicals because remember that if we're finding the radical of a product A B is the same thing as the radical of A multiplied by the radical of B. So using that idea, we can rewrite our expression as four multiplied by the square of 25 multiplied by the skirt of negative one plus two, multiplied by the skirt of 81 multiplied by the square of negative one. No, by definition, we can replace the square of negative one with I. And we can also work out the square roots of 25 and 81. Since they are square numbers, the square of 25 is five. So that's four multiplied by five multiplied by I and the square of 81 is nine. So that's two multiplied by nine, multiplied by I four multiplied by five is 20. So that's 20 I and two multiplied by nine is 18 and 20 I plus 18 and I equals 38. I, no, my answer is in standard form because even though I don't have a real part, I still have an imaginary part which tells us that our answer would be answer choice B 38 I 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. _ _ 5√−16 + 3√−81

Key Concepts

Imaginary Numbers

Standard Form of Complex Numbers

Operations with Complex Numbers

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