Rationalize the denominator and simplify:(\sqrt(2+\sqrt(3)))/(\root(z)(6))
To rationalize the denominator, we want to get rid of the square root in the denominator.
Given: \(\frac{\sqrt{2+\sqrt{3}}}{\sqrt[4]{6}}\)
First, let's simplify the numerator:
\(\sqrt{2 + sqrt{3}} = \sqrt{2 + \sqrt{3}}\)
To rationalize the denominator, we can rewrite \(\sqrt[4]{6}\) as \(\sqrt[4]{6} = (\sqrt{6})^{1/4}\)
Therefore, our expression becomes:
\(\frac{\sqrt{2 + sqrt{3}}}{(\sqrt{6})^{1/4}}\)
Now, we can multiply the numerator and denominator by \((\sqrt{6})^{3/4}\) to rationalize the denominator:
\(\frac{\sqrt{2 + sqrt{3}} \cdot (\sqrt{6})^{3/4}}{(\sqrt{6})^{1/4} \cdot (\sqrt{6})^{3/4}} = \frac{\sqrt{2 + sqrt{3}} \cdot (\sqrt{6})^{3/4}}{(\sqrt{6})}\)
Simplify the expression:
\(\frac{\sqrt{2 + sqrt{3}} \cdot (\sqrt{6})^{3/4}}{(\sqrt{6})} = \frac{(\sqrt{2 + sqrt{3})(3 \sqrt{6})}{\sqrt{6}} = 3\sqrt{2 + \sqrt{3}}\)
Therefore, the rationalized and simplified expression is \(3\sqrt{2 + \sqrt{3}}\)