For Individual or Group Work (Exercises 147 – 150)In calculus, it...

Question

For Individual or Group Work (Exercises 147 – 150)In calculus, it is sometimes desirable to rationalize a numerator. To do this, we multiply the numerator and the denominator by the conjugate of the numerator. For example,

6 - √2 = 6 - √2 • 6 + √2 = 36 - 2 = 34 = 17 .
 4 4 6 + √2 4(6 + √2) 4(6 + √2) 2(6 + √2)

Rationalize each numerator.

2√10 + √7
 30

Accepted Answer

Welcome back. Everyone in this problem. We want to rationalize the numerator of the caution. Five multiplied by the spirit of 13 plus the spirit of all divided by eight. For our answer choices, A is four divided by open bracket. 157 open parenthesis, five multiplied by the square root of 13 plus the square root of closed parentheses. Closed bracket. B is four divided by open bracket. 157 open parenthesis, five multiplied by the square of 13 minus the square root of 11 closed parentheses. Closed bracket. C is 157 divided by open bracket. Four multiplied by open parentheses. Five multiplied by the square of 13 minus the square of 11 closed parentheses closed bracket. And finally D is 157 divided by open bracket. Four multiplied by open parentheses. Five multiplied by the spirit of 13 minus the skirt of closed parentheses, close bracket. Now, what do we already know? Well, it says that we want to rationalize the numerator and that simply means that we want to multiply our numerator, the radical sum by some value that will get rid of the radicals no, how can we get rid of the radicals from a radical? So we'll recall that we can multiply it by its conjugate. And recall that for a radical sum, it is written in the form A plus the square of B, then its conjugate would be equal to A minus the squared of B. In other words, it keeps the same integer part but uses the opposite radical part. Now granted in our expression, the integer part is a part of the product with a radical. But we can still use the same idea for the conjugate because no four or numerator five multiplied by the square of 13 plus the square root of 11. Using the same idea. It's conjugate would be equal to five multiplied by the script of minus the square root of 11. Know that we have that conjugate, we can go ahead and multiply our numerator by it. So we're going to have five multiplied by the square of 13 plus the square of 11 all divided by eight. And given, we are multiplying the numerator by the conjugate. We'll also have to multiply the denominator by the conjugate because that helps to keep the same value of the expression. So let's go ahead and do that notice that the numerators that of both cautions represent the difference of two squares. Again, recall that if I have the sum, if I'm multiplying the sum and the difference of two terms, it's the same thing as multiplying it at the, as the difference of both squares. So I can rewrite my numerator as five multiplied by the square of 13 all squared minus the square of 11 all squared. And that's the whole point of using the conjugate. No, because we're squaring the radicals, it will help to get rid of the square root. Likewise, in our denominator, we're not going to have eight all multiplied by five multiplied by the square of 13 minus the square root of 11. Now let's continue working. Five multiplied by the square root of 13 squared. Well, let's see. Notice that that means we're going to square both terms. So that term five multiplied by the square root of 13 squared equals five squared multiplied by the square root of 13 squared, where five squared is 25 the square of 13 squared is and 25 multiplied by 13 equals 325. So I can write rock 325 here and the square root of 11 squared equals in our denominator eight multiplied by five multiplied by the square of 13. Well, I'm not going to do anything to it for no, because I'm going to see if there's some way to factor out that eight from my numerator. No, 325 minus 11 equals 314 and all of that. And well, the 314 is going to be divided by eight multiplied by five multiplied by the square of 13 minus the square of 11. And now let's think, are there any common terms? Any factors here? Well, yes, two is a factor of 314 and eight, two into 314 goes times and two into eight goes four times. So by simplifying, I can now write this as 157 all divided by four multiplied by five, multiplied by the square of 13 minus the square root of 11. That would have been my final answer after rationalizing. And when we go back to our answer choices, that would have been answer choice. D thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Rationalization

Conjugates

Simplifying Radicals

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