Rationalize each denominator. See Example 8.

√2 - √3
————
√6 -...

Question

Rationalize each denominator. See Example 8.

√2 - √3
————
 √6 - √5

Accepted Answer

Welcome back. Everyone. In this problem, we want to rationalize the denominator of the expression of the square root of 19 minus the square root of seven, all divided by the square of 11 plus the square root of 10. For our answer choices, A is the spirit of 209 minus the spirit of 190 minus the spirit of 77 minus the square root of 70 B is a skirt of minus the skirt of 190 plus seven. The skirt of 77 plus the square root of 70 C is a skirt of 209 minus the square root of 190 minus the skirt of 77 plus a skirt of 70 D is a skirt of 190 minus the square root of minus the square of 77 plus the square root of 70. Now, what do we want to do here? We want to rationalize the denominator of our radical expression and our expression is a square of 19 minus the square of seven divided by the square of 11 plus the square root of to rationalize means that we want to get rid of the radical oath of the denominator. And given our radical or denominator is a radical sum, then we can get rid of the radical when we multiply by the conjugate. What does that mean? Well, it just means that if we have a radical sum. So let's recall that to find a conjugate of a radical sum. Let's write it in the form A plus A squared of B means that we keep our integer part A but we make our radical part negative or the opposite of the radical part. So given that, that's the definition of our conjugate, then that tells us that the conjugate of our denominator, which is the square level plus the square root of 10, even though this is not an integer part. The first term isn't an integer. We can use the same idea. So we keep the first part but we make the second value or our radical negative. So the conjugate of the square of 11 plus the square of 10 equals the square of 11 minus the square root of 10. So now we can multiply our expression by the conjugate in the numerator and in the denominator. So that's the skirt of 19 minus the skirt of seven divided by the skirt of 11 plus the skirt of 10, multiplied by the skirt of 11 minus the skirt of 10, divided by the skirt of 11 minus the square of 10. And we put it in the numerator as well to preserve the value of our expression because essentially we're just multiplying by one. Now, when we go ahead and do that in our numerator, we'd have the product of the square of 19 minus the square root of seven and the square root of 11 minus the square root of 10. In our denominator, we're going to have the squared of 11 plus the skirt of 10 multiplied by the skirt of 11 minus the square of 10. Now notice that these are the sums and this is the sum and the difference of the same set of terms. And to help us to simplify that we can use the property of the difference of two squares because recall that if we are multiplying the sum and the difference of two terms, then we can rewrite it as the difference of both of those terms squares. So that tells us that in our denominator, we can rewrite it as the square root of 11 squared minus the square root of 10 squared. OK. So let's go ahead now and simplify first, we'll have the spirit of 19 multiplied by the square of 11. Again, to make this simpler, what can help us is to recall what we know about the product of radicals. Because remember that if we are multiplying two radicals, the spirit of a multiplied by the spirit of B is the same thing as finding the radical of their product. In this case, A B. So again, we can apply this product to help us because no, we're going to have the skirt of 19 multiplied by the skirt of 11, which would be the same thing as the skirt of 19 multiplied by 11, which would have been 209. Next, we're going to have the square of 19 multiplied by a negative square root of 10, which that would be minus the square root of 190. Afterwards, the square of ele sorry, the negative spirit of seven multiplied by the spirit of seven as a negative square root of 77 and the negative square of seven multiplied by the negative square root of 10 is a positive square root of 70. Now all of that will be divided by the square of the spirit of 11, which is just 11 and the square of the spirit of 10, which would be 10. Now 11 minus 10, that equals one. So that tells us our answer would be the square of 209 minus the square root of 190 minus the square root of plus the square of 70 all divided by one, which would have been this value that we have right here. Now, when we look at on our answer choices, that would have been answer choice. C thanks a lot for watching everyone. I hope this video helped.

Rationalize each denominator. See Example 8. √2 - √3 ———— √6 - √5

Key Concepts

Rationalizing the Denominator

Conjugates

Properties of Square Roots

Watch next