Question

1. \\(\frac{\sqrt{3} - \sqrt{5}}{4\sqrt{19}}\\) \\(\frac{\sqrt{19}}{\sqrt{19}}\\)

Explanation:

Step1: Rationalize the denominator

To rationalize the denominator of \(\frac{\sqrt{3} - \sqrt{5}}{4\sqrt{19}}\), we multiply the numerator and the denominator by \(\sqrt{19}\).

$$ \frac{(\sqrt{3} - \sqrt{5})\times\sqrt{19}}{4\sqrt{19}\times\sqrt{19}} $$

Step2: Simplify the denominator

Simplify the denominator using the property \(\sqrt{a}\times\sqrt{a}=a\) (for \(a\geq0\)). So \(\sqrt{19}\times\sqrt{19} = 19\), and the denominator becomes \(4\times19=76\).

$$ \frac{(\sqrt{3} - \sqrt{5})\sqrt{19}}{76} $$

Step3: Distribute the \(\sqrt{19}\) in the numerator

Using the distributive property \(a(b - c)=ab - ac\) (here \(a = \sqrt{19}\), \(b=\sqrt{3}\), \(c = \sqrt{5}\)), we get:

$$ \frac{\sqrt{3\times19}-\sqrt{5\times19}}{76}=\frac{\sqrt{57}-\sqrt{95}}{76} $$

Answer:

\(\frac{\sqrt{57}-\sqrt{95}}{76}\)