In Exercises 50 - 53, rationalize the denominator. 14/(√7 - √5)

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Hello. Today we're going to be rationalizing and expression. So the given expression is 25 over the square root of 2 - the square root of three. Now in order to rationalize this quantity, we need to go ahead and multiply by the conjugate of the denominator. Multiplying by the conjugate of the denominator means that we're taking the denominator and multiplying by its opposite. So instead of having a minus sign we're going to be using a plus sign. So the conjugate of the denominator is going to be the square root of two plus the square root of three. All over the square root of two plus the square root of three Combining this together is going to give me 25 times the square root of two plus the square root of three. And all of that is going to be over The square root of two minus the square root of three times the square root of two plus the square root of three. Now, in order to continue simplifying this, we need to go ahead and foil the denominator. That means that we're going to take the square root of two and multiply it to the square root of two. Doing so is going to give me the square root of two squared which is just equal to two. Next we're gonna go ahead and take the square root of two and multiply it to the square root of three. That's going to give me positive square root of two times square root of three. Next we're gonna go ahead and take negative square root of three and multiply to the square root of two. Doing so is going to give me negative square root of two times the square root of three. And finally we're going to go ahead and take the negative square root of three and multiply to the positive square root of three. Now doing so is going to give me negative square root of three squared which is just equal to negative three. So we're gonna go ahead and include that here. Now if you notice we have a positive square root of two times square root of three and a negative square root of two times square root of three, that means that these two quantities are going to cancel each other out. That's then gonna leave us with 2 -3 which is just equal to negative one. So going back to the top here now our denominator is simplified to just negative one. So we're left with 25 times the square root of two plus the square root of three. All over negative one. And negative one just means that we're gonna in the denominator just means we're going to divide the entire thing by negative one. So doing so is just going to give me negative times the square root of two plus the square root of three. So this will be our solution here which means that the solution to this problem is going to be d So I hope this helps you guys work with the conjugate of the denominator, and I'll go ahead and see you all in the next video.

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