Where to get help? Rationalizing a denominator.

[BlueIvy]

Hello, I'm wondering where I can get practice questions with answers for questions like these:

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\(\displaystyle \dfrac{\sqrt{\strut 3\,}\, -\, \sqrt{\strut 2\,}}{\sqrt{\strut 3\,}\, +\, \sqrt{\strut 2\,}}\)

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I've been trying all day to find what I need but it's been very difficult and taking hours. I tried multiplying by the conjugate but I'm not getting my answer.

Any help is appreciated. Thank you.

 

[stapel]

\(\displaystyle \dfrac{\sqrt{\strut 3\,}\, -\, \sqrt{\strut 2\,}}{\sqrt{\strut 3\,}\, +\, \sqrt{\strut 2\,}}\)

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I've been trying all day.... I tried multiplying by the conjugate but I'm not getting my answer.

Multiplying by the conjugate of the denominator is exactly the method that you should be using. Please reply showing your steps and your result, so we can try to find where things are going wrong. Thank you!

 

[BlueIvy]

My working:

. . . . .\(\displaystyle \dfrac{\sqrt{\strut 3\,}\, -\, \sqrt{\strut 2\,}}{\sqrt{\strut 3\,}\, +\, \sqrt{\strut 2\,}}\, \cdot\, \dfrac{\sqrt{\strut 3\,}\, -\, \sqrt{\strut 2\,}}{\sqrt{\strut 3\,}\, -\, \sqrt{\strut 2\,}}\, =\, \dfrac{\left(\sqrt{\strut 3\,}\right)^2\, -\, \left(\sqrt{\strut 2\,}\right)^2}{\left(\sqrt{\strut 3\,}\right)^2\, -\, \left(\sqrt{\strut 2\,}\right)^2}\, =\, \dfrac{3\, -\, 2}{3\, -\, 2}\). . .X

I thought I was supposed to remove the square root once it's squared? Sorry, I know it's a simple question and I was ok with some of the other questions but not this one. Thanks for your help!

 

[stapel]

My working:

. . . . .\(\displaystyle \dfrac{\sqrt{\strut 3\,}\, -\, \sqrt{\strut 2\,}}{\sqrt{\strut 3\,}\, +\, \sqrt{\strut 2\,}}\, \cdot\, \dfrac{\sqrt{\strut 3\,}\, -\, \sqrt{\strut 2\,}}{\sqrt{\strut 3\,}\, -\, \sqrt{\strut 2\,}}\, =\, \dfrac{\left(\sqrt{\strut 3\,}\right)^2\, -\, \left(\sqrt{\strut 2\,}\right)^2}{\left(\sqrt{\strut 3\,}\right)^2\, -\, \left(\sqrt{\strut 2\,}\right)^2}\,...\)

How did you get that the two products (top and bottom, with different factors) resulted in the same expressions? In particular, how did you get that this:

. . . . .\(\displaystyle \left(\sqrt{\strut 3\,}\, -\, \sqrt{\strut 2\,}\right)\, \left(\sqrt{\strut 3\,}\, -\, \sqrt{\strut 2\,}\right)\)

...equalled anything other than this?

. . . . .\(\displaystyle \sqrt{\strut 3\,}\, \sqrt{\strut 3\,}\, +\, \left(-\sqrt{\strut 2\,}\, \sqrt{\strut 3\,}\right)\, +\, \left(-\sqrt{\strut 3\,}\, \sqrt{\strut 2\,}\right)\, +\, \left(-\sqrt{\strut 2\,}\right)\, \left(-\sqrt{\strut 2\,}\right)\)

. . . . .\(\displaystyle \left(\sqrt{\strut 3\,}\right)^2\, -\, 2\, \sqrt{\strut 3\,}\, \sqrt{\strut 2\,}\, +\, \left(-\sqrt{\strut 2\,}\right)^2\)

When you reply, please show all the steps in your multiplications, etc. Thank you!

 

[BlueIvy]

Thank you both