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 everyone. In this problem, we want to simplify the product of the square root of negative 20. And the expression, the square root of negative six minus the square root of seven. And we want to express our answer in standard form if necessary for our answer choices. A is two multiplied by the spirit of 30 plus two, multiplied by the spirit of 35 IB is two multiplied by the spirit of 30 minus two, multiplied by the spirit of 35 I C is two multiplied by the skirt of 35 plus two, multiplied by the square root of 30 I and D is two multiplied by the skirt of 35 minus two, multiplied by the square root of 30 I. Now, what are we trying to do here? Well, we want to take this, this expression and simplify it, writing or answering Standard form. And this is talking about the standard form for complex numbers. Now it uses that because notice we have terms that we're finding the square root of a negative number. And by definition, that's what a complex number is. Because recall that a complex number, the complex unit I is actually equal to the square root of negative one. And that means if we're gonna write it in standard form, we are going to write our number in the form A plus B I where A is the real part and B is the imaginary part. So let's go ahead multiply our expressions and then see how we can get them to simplify. Now by distributing I'm going to multiply the square root of negative 20 by the square root of negative six and the square of negative 20 by nega the negative of the square root of seven that is minus the square of negative 20 multiplied by the square root of seven. Now we can rewrite each of our products in a simpler form because if you recall the product rule of radicals, remember that if we are multiplying two radicals squared of A by the square of B, it's the same thing as the square root of their product. That is the square root of A B. So by applying that property here, we can take it a step further to rewrite this as the square root of negative 20 multiplied by negative six minus the square root of negative 20 multiplied by seven by simplifying negative 20 multiplied by negative six is positive 120 negative 20 multiplied by seven is negative 140. No, we can continue to simplify if we were to find the square factors in our product because our radical is not in its simplest form. It's actually in the simplest form when we remove us, when there are no more square factors for the number. So let's think about it does 120 have any square factors. Well, it does. So using the same property where four is a factor of 120 we can rewrite 120 as four multiplied by 30. Likewise, we can rewrite negative 140 as four multiplied by negative 35. No, by doing that. And using the same product property of that we used before are our answer or, or, or expression that we're working on? I was a little lost for words there, but we can rewrite it now as the script of four multiplied by the script of minus the script of four multiplied by the square root of negative 35. The square of four is two. So I can write this as two multiplied by the square of 30 minus two multiplied by the square root of negative 35. No, we're almost there. Remember we said that we want our answer in standard form. So the question is how can I convert this or how can I turn this in the standard form? Again, let me use the definition of a complex number. A complex number is are the unit of a complex number equals the spirit of negative one. So I can rewrite the radical in my second expression that is the spirit of negative 35 as the spirit of 35 multiplied by negative one. No, I have my imaginary unit. And once I do that, I can rewrite this as two multiplied by the skirt of 35 multiplied by the skit of negative one, which in its simplest form is not two multiplied by the square root of 35. I therefore, my final answer is two multiplied by the square of 30 minus two, multiplied by the square root of 35. I. And when we look in our answer choices that would have been answer choice. B 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 (√−3 − √5 )

Key Concepts

Complex Numbers

Square Roots of Negative Numbers

Standard Form of Complex Numbers

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