\[\frac{\sqrt{3}}{\sqrt{7+\sqrt{5}}}\]
1
Rationalize the denominator: 37+5⋅7+57+5=37+57+5\frac{\sqrt{3}}{\sqrt{7+\sqrt{5}}} \cdot \frac{\sqrt{7+\sqrt{5}}}{\sqrt{7+\sqrt{5}}}=\frac{\sqrt{3}\sqrt{7+\sqrt{5}}}{7+\sqrt{5}}.
37+57+5\frac{\sqrt{3}\sqrt{7+\sqrt{5}}}{7+\sqrt{5}}

2
Rationalize the denominator: 37+57+5⋅7−57−5=37+5(7−5)72−52\frac{\sqrt{3}\sqrt{7+\sqrt{5}}}{7+\sqrt{5}} \cdot \frac{7-\sqrt{5}}{7-\sqrt{5}}=\frac{\sqrt{3}\sqrt{7+\sqrt{5}}(7-\sqrt{5})}{{7}^{2}-{\sqrt{5}}^{2}}.
37+5(7−5)72−52\frac{\sqrt{3}\sqrt{7+\sqrt{5}}(7-\sqrt{5})}{{7}^{2}-{\sqrt{5}}^{2}}

3
Simplify  72{7}^{2}  to  4949.
37+5(7−5)49−52\frac{\sqrt{3}\sqrt{7+\sqrt{5}}(7-\sqrt{5})}{49-{\sqrt{5}}^{2}}

4
Use this rule: x2=x{\sqrt{x}}^{2}=x.
37+5(7−5)49−5\frac{\sqrt{3}\sqrt{7+\sqrt{5}}(7-\sqrt{5})}{49-5}

5
Simplify  49−549-5  to  4444.
37+5(7−5)44\frac{\sqrt{3}\sqrt{7+\sqrt{5}}(7-\sqrt{5})}{44}

Done

Decimal Form: 0.569924
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