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. (3√-5)(- 4√-12)

Accepted Answer

Welcome back. I'm so glad you're here. We've got this expression, I'll rewrite it. We've got two times the square root of negative seven Times -5 sq Root of -18. And we're to do a couple of things here, were to simplify it and were to write our answer in standard form. Well, what is the standard form here? We've got the square root of negative numbers, which means we're dealing with complex numbers and we're going to write that in a plus B I. Standard form. And now when we note that we have the square root of a negative number, the first step of our simplification process is to rewrite this. In terms of that I for our standard form, we recall that the square root of negative one equals I. And this two times the square root of negative seven could be rewritten as two times the square root of negative one, times seven. And over here this negative five Scruton of negative 18 could be rewritten as negative one times 18 inside that square root. And this is great because we just said, the square root of negative one is I, which means we can pull out This square root of negative one for both terms. And now we've got two, i times the square root of seven and we've got negative five, I times the square root of 18. Great, let's see if we can do any further simplification inside those square roots before we start multiplying and we take a look here at 18 because 77 is just a prime number. We're not going to get that factored down at all. But 18 we could build a factor tree for 18, 18 is the same as six times three and six is two times three. So that's two times three squared. Great will bring the rest of this down to I squared of seven times negative five I. And inside here is now two times three squared. And we know that the square root of three squared. Well that can come on out. So now we've got negative five times three I squared of two And bring down our two I square to seven. And we'll just multiply our negative five times three. So now we bring down to I squared of seven and we've got negative 15 I squared of two. And our square square roots are looking pretty good here. So now let's just multiply the parts that we can multiply this two. I gets multiplied by the negative 15 I and that gives us a negative 30 I squared. And likewise this seven inside the square root gets multiplied by the two inside of the square root and we've got the square root of 14. This is looking good. We recall from previous lessons that I squared by its very definition is negative one. So instead of negative 30 I squared, we can rewrite that as negative 30 times negative one squared 2, 14 and negative 30 times negative one. That's just 30. So we've got 30 square root 14. We can't simplify that anymore. So we look at our answer choices and that matches with answer choice. C. 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. (3√-5)(- 4√-12)

Key Concepts

Complex Numbers

Multiplication of Complex Numbers

Standard Form of Complex Numbers

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