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.
           ___
−8 + √−32   / 24

Accepted Answer

Welcome back everyone. In this problem, we want to simplify the expression negative six plus the square root of negative 27 all divided by 15. And we want to express our answer in standard form if necessary for our answer choices, A is six fifths plus three multiplied by the square of three divided by five I B is six fifths minus three, multiplied by the skirt of three divided by five IC is negative 2/5 minus the skirt of three divided by five I and D is negative 2/5 plus the skirt of three divided by five. I. Now, in simplifying our expression, we want to write 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 it in a form A plus B I where A is the real part and B is the imaginary part. And the reason we want to write it in standard form is because we have the square root of a negative number involved here. And by definition, that's the unit of an imaginary number because also recall that an imaginary unit I equals the square root of negative one. So what we're planning to do here is to simplify our expression until we can get it written in this form. Now, before we do that, we want to have a radical in its simplest form. And right. No, it's not because 27 has a square factor. So if we can factor out that square number, then we can make our problem easier to solve. So let's think about it. What is a square factor of 27? Well, if we think about it, we can rewrite 27 as nine multiplied by three. So I can rewrite my problem as nine multiplied by negative three because it was negative 27 and I can right, all of that divided by 15. No, another property that can help us here is the product rule for radicals and recall that what it tells us is that if we have 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 property, we can rewrite our numerator as negative six plus the square root of nine multiplied by the square root of negative three, all divided by 15. Now the square root of nine is three. So my problems is known negative six plus three multiplied by the square root of negative three divide all divided by 15. It's still not in the standard form though. So we can use our definition for a complex number to break down the spirit of negative three. Because using that property, we can rewrite it as negative six plus three multiplied by the spirit of three multiplied by negative one and no, the square root of negative one. Let me come over to the come over to the right. Sometimes I mix up my sides, we can rewrite that as negative six plus three multiplied by the square of three multiplied by the skirt of negative one all divided by 15. But again, remember the square of negative one is I so no, that can be written as negative six plus three multiplied by the square of three I and all of that is divided by 15. Finally, we cannot div divide through our expression by 15 and negative six divided by 15 plus three multiplied by the square root of three I divided by 15 by factoring out. Notice three is a common factor of six and 15. 3 into negative six goes negative two times three into 15, 5 times. And for a second term three is a common factor of three and 15. 3 goes into 313 goes into 15, 5 times. So I can now rewrite my expression as negative 2/5 plus the square root of three divided by five I. And that would be my final answer because no, it is in the standard form where negative 2/5 is my real part and the square root of three divided by five. I is my imaginary part. When we go up to 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. ___ −8 + √−32 / 24

Key Concepts

Complex Numbers

Standard Form of Complex Numbers

Simplifying Square Roots of Negative Numbers

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