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 (√-3 - √5)

Accepted Answer

Welcome back. I'm so glad you're here. We've got this expression, I'll rewrite it. We've got the square root of negative 12 times parentheses. The square root of negative five minus square root of seven and that one is not negative. And we are to rewrite this answer in standard form, what does that mean? Well we've got some square roots of negative numbers, meaning we're dealing with complex numbers here. And the standard form for that is A plus B. I. And now the first step in our simplification process here is when we see that square root of a negative number, we're going to rewrite that in terms of I. So we recall from previous lessons that I is the square root of negative one and we could rewrite the ones that have a negative inside as well. This squared of negative 12 is negative square root of negative one times 12. And this squirt of -5 is the same as the square root of -1 times five. And then that's minus the square root of seven. And now we can just take this squared of negative one and we can pull that out as an eye. So now we've got I square root of times I squared of five minus the square root of seven. Great. This is looking better. Now we want to take a look at our terms inside the square roots and say, can we simplify these, Can we bring anything out? Well seven and five. Those are prime numbers were not going to get those down but this let's take a look at what we can bring out, build a factor tree for 12, 12 is four times three and four is two times two. So this is the same as two squared times three. So now we've got I times the square root of two squared times three. All times I squared to five minus the square root of seven. And inside that square root bracket, this two squared is really excited because the squared of two squared. Well that is just too. So now we've got two I times the square root of three. All times I squared of five minus the square root of seven. And all those square brackets are now as simplified as they're going to get. So we can now distribute this to I squared of three times the two terms inside of our parentheses. So what does that look like to I squared of three times I squared to five. You take the two I times the eye gives us two I squared. And inside the square roots three times five is 15. Now we've got two I times a negative one. So that's gonna be a minus two I. And inside the square roots that's the squared of three times the square root of seven, which gives us the square root of 21. Well this is looking pretty good. But now we recall from previous lessons that I squared by, its very definition is negative one. We've got this, I squared here. So that could be rewritten as two times not I squared, but negative one square to 15 minus two I squared of 21. And now we can just go ahead and do two times negative one, which is negative two and bring down our square root of 15 minus two, I square root of 21. And that is as simplified as this one is going to get. So we look at our answer choices and this matches with answer choice. A well done, we'll catch you on the next one.

In Exercises 37–52, perform the indicated operations and write the result in standard form. √-8 (√-3 - √5)

Key Concepts

Complex Numbers

Square Roots of Negative Numbers

Standard Form of Complex Numbers

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