Question 193955
<font face="Garamond" size="+2">


In the first place, why are you shouting at us?  All CAPS is the electronic communications equivalent of shouting, and is both rude and annoying.


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \frac{\sqrt{10}-2}{\sqrt{10}+3}]


Simplest form requires taking the radical out of the denominator.  Simply squaring the denominator won't do because you would still have a radical term because:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ (a + b)^2 = a^2 + ab + b^2]


However, multiplying the denominator by its conjugate, namely *[tex \LARGE \sqrt{10} - 3], works because that results in the difference of two squares, thus:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ (a + b)(a - b) = a^2 - b^2]


But you can't just multiply the denominator by something without changing the value of the rational expression.  You need to multiply by 1 in the form of *[tex \LARGE \frac{\sqrt{10}-3}{\sqrt{10}-3}], thus:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \left(\frac{\sqrt{10}-2}{\sqrt{10}+3}\right)\left(\frac{\sqrt{10}-3}{\sqrt{10}-3}\right)]


Apply FOIL to the numerator, and the difference of two squares result to the denominator:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \frac{10 - 5\sqrt{10} + 6}{10 - 9} = \frac{16 - 5\sqrt{10}}{1} = 16 - 5\sqrt{10}]



John
*[tex \LARGE e^{i\pi} + 1 = 0]
</font>