Question

simplify by rationalizing the denominator.
\\(\frac{7}{\sqrt{171}}\\)
symbol (\\(\sqrt{\square}\\)), type
oot\

Explanation:

Step1: Simplify the square root in the denominator

First, factorize 171. We know that \(171 = 9\times19\), so \(\sqrt{171}=\sqrt{9\times19}\). Using the property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (where \(a = 9\) and \(b = 19\) and \(a\geq0,b\geq0\)), we get \(\sqrt{9\times19}=\sqrt{9}\times\sqrt{19}=3\sqrt{19}\). So the original fraction becomes \(\frac{7}{3\sqrt{19}}\).

Step2: Rationalize the denominator

To rationalize the denominator, we multiply the numerator and the denominator by \(\sqrt{19}\). So we have \(\frac{7\times\sqrt{19}}{3\sqrt{19}\times\sqrt{19}}\).

Step3: Simplify the denominator

Simplify the denominator: \(\sqrt{19}\times\sqrt{19}=19\) (by the property \(\sqrt{a}\times\sqrt{a}=a\) for \(a\geq0\)). So the fraction becomes \(\frac{7\sqrt{19}}{3\times19}=\frac{7\sqrt{19}}{57}\).

Answer:

\(\frac{7\sqrt{19}}{57}\)