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.

6 - √3
 8

Accepted Answer

Welcome back. Everyone. In this problem, we want to rationalize the numerator eight minus the square of 55 all divided by 17. For our answer choices A is nine divided by open bracket. 17 multiplied by open parenthesis. Eight plus the square of 55 closed pas close bracket B is almost the same thing as A. But we're subtracting in our denominator C is 17 divided by open bracket. Nine, open parentheses, eight plus the skirt of 55 closed parentheses. Close the bracket and D uses the same values. But instead we are subtracting in our denominator. Now, what do we want to find out here? We want to rationalize our new narrator and that means that we, we want, we want to rewrite it without a radical to rationalize a radical sum. We can use the conjugate because recall that for a radical sum in the form A plus the squared of B, then its conjugate OK, would be equal to the same integer part A but the opposite radical part minus the spirit of B OK. So in this case, this tells us that the conjugate of our numerator eight minus the square of 55 would be equal to eight plus the square of 55. So we can multiply our numerator and denominator by that conjugate in order to one preserve the value and to get rid of the radical from our numerator. So let's do that. So this tells us that we can rewrite our expression eight minus the script of 55 all divided by 17 by multiplying it by eight plus the square root of 55 in the numerator and the denominator. No recall as well. That a reason we use the conjugate in the numerator is because it gives us the difference of two squares because notice the numerator and the denominator in the form A plus B multiplied by A minus B, which means that it's a difference of two squares and can be written as A squared minus B squared. And that's the whole point of rationalizing because we can square our square root and get rid of the radical. So we can rewrite it as our numerator as eight squared minus the square root of 55 all squared. And that will be divided by 17, multiplied by eight plus the square root of 55. Now, when we think about it, it's squared equals 64 and the square root of 55 are squared equals and 64 minus the square, sorry, 64 minus 55 equals nine. So our question now becomes nine divided by 17, multiplied by eight plus the square root of 55. When we think about our answer choices or when we look back on our answer choices, notice that that would have been answer choice. A thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Rationalization

Conjugate

Simplifying Expressions

Watch next