This is driving me insane, I must be doing the arithmetic wrong. I just have to solve the equation (1), but I don't think I get the right answer
You must convert $\displaystyle \sqrt{3} $ and $\displaystyle \sqrt{7} $ to fraction so that you can summing with the fraction $\displaystyle \frac{1}{\sqrt{3} + \sqrt{7}} $. Try to figure out which factor you can multiply $\displaystyle \sqrt{3} $ and $\displaystyle \sqrt{7} $ with so you'll get a denominator which is equal to this one $\displaystyle \sqrt{3} + \sqrt{7} $.
You say that you're "solving an equation", but there is no variable. In what you've posted, one line does not follow from the previous (such as where denominators disappear and then later reappear).
Are the various lines different exercises and you need to "simplify"...? Assuming so, I think the following is what you did, with the missing steps filled in:
. . . . .$\displaystyle \mbox{Simplify: }\, \sqrt{3}\, +\, \sqrt{7}\, +\, \frac{1}{\sqrt{3}\, +\, \sqrt{7}}$
. . . . .$\displaystyle \mbox{My work: }\, \sqrt{3}\, +\, \sqrt{7}\, +\, \left(\frac{1}{\sqrt{3}\, +\, \sqrt{7}}\right)\left(\frac{\sqrt{3}\, -\, \sqrt{7}}{\sqrt{3}\, -\, \sqrt{7}}\right)$
. . . . .$\displaystyle \mbox{rationalize: }\,\sqrt{3}\, +\, \sqrt{7}\, +\, \frac{\sqrt{3}\, -\, \sqrt{7}}{3\, -\, 7}$
. . . . .$\displaystyle \mbox{simplify: }\,\sqrt{3}\, +\, \sqrt{7}\, +\, \frac{\sqrt{7}\, -\, \sqrt{3}}{4}$
. . . . .$\displaystyle \mbox{common denominator: }\,\frac{4\sqrt{3}}{4}\, +\, \frac{4\sqrt{7}}{4}\, +\, \frac{\sqrt{7}\, -\, \sqrt{3}}{4}$
But I'm afraid I can't figure out how you arrived at your later lines...?