How to simply this fraction with irrational denominators? 1 1 +

How to simply this fraction with irrational denominators?
$\frac{1}{1+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{5}}+\frac{1}{\sqrt{5}+\sqrt{7}}\frac{1}{\sqrt{7}+3}$
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Mateo Carson
If it is $\frac{1}{1+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{5}}+\frac{1}{\sqrt{5}+\sqrt{7}}+\frac{1}{\sqrt{7}+3}$ then
$=\frac{1}{1+\sqrt{3}}\frac{1-\sqrt{3}}{1-\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{5}}\frac{\sqrt{3}-\sqrt{5}}{\sqrt{3}-\sqrt{5}}+\frac{1}{\sqrt{5}+\sqrt{7}}\frac{\sqrt{5}-\sqrt{7}}{\sqrt{5}-\sqrt{7}}+\frac{1}{\sqrt{7}+3}\frac{\sqrt{7}-3}{\sqrt{7}-3}$
$=\frac{1-\sqrt{3}}{1-3}+\frac{\sqrt{3}-\sqrt{5}}{3-5}+\frac{\sqrt{5}-\sqrt{7}}{5-7}+\frac{\sqrt{7}-3}{7-9}$
$=\frac{1-\sqrt{3}}{-2}+\frac{\sqrt{3}-\sqrt{5}}{-2}+\frac{\sqrt{5}-\sqrt{7}}{-2}+\frac{\sqrt{7}-3}{-2}$
$=-\frac{1}{2}\left(1-\sqrt{3}+\sqrt{3}-\sqrt{5}+\sqrt{5}-\sqrt{7}+\sqrt{7}-3\right)$
$=-\frac{1}{2}\left(1-3\right)$
$=-\frac{1}{2}×-2$
$=1$
Did you like this example?
Raul Walker
Hint. If you rationalise the denominator of each term, you always obtain the same denominator and, in the numerator, a telescoping sum.