simplify [1/root3+root2] + [1/root2+1]
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![\quad \: <br />\frac{1}{ \sqrt{3} + \sqrt{2} } + \frac{1}{ \sqrt{2} + 1} \\ \\ <br />= \frac{1( \sqrt{3} - \sqrt{2} )}{( \sqrt{3} + \sqrt{2} )( \sqrt{3} - \sqrt{2} )} + \frac{1( \sqrt{2} - 1)}{( \sqrt{2} + 1)( \sqrt{2} - 1)} \sf ---------[Rationalising] \\ \\ <br />= \frac{ \sqrt{3} - \sqrt{2} }{( { \sqrt{3} )}^{2} - {( \sqrt{2} )}^{2} } + \frac{ \sqrt{2} - 1 }{ {( \sqrt{2} )}^{2} - {(1)}^{2} } --------[ \because ( {a}^{2} - {b}^{2} ) = (a-b)(a+b) ]\\ \\ <br />= \frac{ \sqrt{3} - \sqrt{2} }{3 - 2} + \frac{ \sqrt{2} - 1}{2 - 1} \\ \\ <br />= \frac{ \sqrt{3} - \sqrt{2} }{1} + \frac{ \sqrt{2} - 1}{1} \\ \\ <br />= \sqrt{3} - \cancel{\sqrt{2} } + \cancel{ \sqrt{2} } - 1 \\ \\ \bf <br />= \quad \sqrt{3} - 1 \quad \: <br />\frac{1}{ \sqrt{3} + \sqrt{2} } + \frac{1}{ \sqrt{2} + 1} \\ \\ <br />= \frac{1( \sqrt{3} - \sqrt{2} )}{( \sqrt{3} + \sqrt{2} )( \sqrt{3} - \sqrt{2} )} + \frac{1( \sqrt{2} - 1)}{( \sqrt{2} + 1)( \sqrt{2} - 1)} \sf ---------[Rationalising] \\ \\ <br />= \frac{ \sqrt{3} - \sqrt{2} }{( { \sqrt{3} )}^{2} - {( \sqrt{2} )}^{2} } + \frac{ \sqrt{2} - 1 }{ {( \sqrt{2} )}^{2} - {(1)}^{2} } --------[ \because ( {a}^{2} - {b}^{2} ) = (a-b)(a+b) ]\\ \\ <br />= \frac{ \sqrt{3} - \sqrt{2} }{3 - 2} + \frac{ \sqrt{2} - 1}{2 - 1} \\ \\ <br />= \frac{ \sqrt{3} - \sqrt{2} }{1} + \frac{ \sqrt{2} - 1}{1} \\ \\ <br />= \sqrt{3} - \cancel{\sqrt{2} } + \cancel{ \sqrt{2} } - 1 \\ \\ \bf <br />= \quad \sqrt{3} - 1](https://tex.z-dn.net/?f=+%5Cquad+%5C%3A+%3Cbr+%2F%3E%5Cfrac%7B1%7D%7B+%5Csqrt%7B3%7D+%2B+%5Csqrt%7B2%7D+%7D+%2B+%5Cfrac%7B1%7D%7B+%5Csqrt%7B2%7D+%2B+1%7D+%5C%5C+%5C%5C+%3Cbr+%2F%3E%3D+%5Cfrac%7B1%28+%5Csqrt%7B3%7D+-+%5Csqrt%7B2%7D+%29%7D%7B%28+%5Csqrt%7B3%7D+%2B+%5Csqrt%7B2%7D+%29%28+%5Csqrt%7B3%7D+-+%5Csqrt%7B2%7D+%29%7D+%2B+%5Cfrac%7B1%28+%5Csqrt%7B2%7D+-+1%29%7D%7B%28+%5Csqrt%7B2%7D+%2B+1%29%28+%5Csqrt%7B2%7D+-+1%29%7D+%5Csf+---------%5BRationalising%5D+%5C%5C+%5C%5C+%3Cbr+%2F%3E%3D+%5Cfrac%7B+%5Csqrt%7B3%7D+-+%5Csqrt%7B2%7D+%7D%7B%28+%7B+%5Csqrt%7B3%7D+%29%7D%5E%7B2%7D+-+%7B%28+%5Csqrt%7B2%7D+%29%7D%5E%7B2%7D+%7D+%2B+%5Cfrac%7B+%5Csqrt%7B2%7D+-+1+%7D%7B+%7B%28+%5Csqrt%7B2%7D+%29%7D%5E%7B2%7D+-+%7B%281%29%7D%5E%7B2%7D+%7D+--------%5B+%5Cbecause+%28+%7Ba%7D%5E%7B2%7D+-+%7Bb%7D%5E%7B2%7D+%29+%3D+%28a-b%29%28a%2Bb%29+%5D%5C%5C+%5C%5C+%3Cbr+%2F%3E%3D+%5Cfrac%7B+%5Csqrt%7B3%7D+-+%5Csqrt%7B2%7D+%7D%7B3+-+2%7D+%2B+%5Cfrac%7B+%5Csqrt%7B2%7D+-+1%7D%7B2+-+1%7D+%5C%5C+%5C%5C+%3Cbr+%2F%3E%3D+%5Cfrac%7B+%5Csqrt%7B3%7D+-+%5Csqrt%7B2%7D+%7D%7B1%7D+%2B+%5Cfrac%7B+%5Csqrt%7B2%7D+-+1%7D%7B1%7D+%5C%5C+%5C%5C+%3Cbr+%2F%3E%3D+%5Csqrt%7B3%7D+-+%5Ccancel%7B%5Csqrt%7B2%7D+%7D+%2B+%5Ccancel%7B+%5Csqrt%7B2%7D+%7D+-+1+%5C%5C+%5C%5C+%5Cbf+%3Cbr+%2F%3E%3D+%5Cquad+%5Csqrt%7B3%7D+-+1)
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