Simplify ![$\frac{3}{\sqrt[5]{16}}+\frac{1}{\sqrt{3}}$](/api/ssl-img-proxy?src=https%3A%2F%2Flatex.artofproblemsolving.com%2Fd%2F4%2F9%2Fd49d076e1e74dfb8e8603510ca2114f81eda673a.png)
and rationalize the denominator. The result can be expressed in the form ![$\frac{a^2\sqrt[5]{b}+b\sqrt{a}}{ab}$](/api/ssl-img-proxy?src=https%3A%2F%2Flatex.artofproblemsolving.com%2Fb%2F0%2F8%2Fb08afd6f93a5c548fbe78c1a12a603aadeeba1e5.png)
, where 
and 
are integers. What is the value of the sum 
?
+9466 \(\quad\,\frac{3}{\sqrt[5]{16}}\,+\,\frac{1}{\sqrt3} \\~\\ =\,\frac{3}{\sqrt[5]{2^4}}\,+\,\frac{1}{\sqrt3} \\~\\ =\,\frac{3}{2^\frac45}\,+\,\frac1{3^\frac12}\\~\\ =\,\frac{3\,\cdot\,2^\frac15}{2^\frac45\,\cdot\,2^\frac15}\,+\,\frac{1\,\cdot\,3^\frac12}{3^\frac12\,\cdot\,3^\frac12}\\~\\ =\,\frac{3\,\cdot\,2^\frac15}{2}\,+\,\frac{3^\frac12}{3}\\~\\ =\,\frac{3\,\cdot\,2^\frac15\,\cdot\,3}{2\,\cdot\,3}\,+\,\frac{3^\frac12\,\cdot\,2}{3\,\cdot\,2}\\~\\ =\,\frac{3^2\,\cdot\,2^\frac15}{3\cdot2}\,+\,\frac{3^\frac12\,\cdot\,2}{3\cdot2}\\~\\ =\,\frac{3^2\,\cdot\,2^\frac15\,+\,3^\frac12\,\cdot\,2}{3\cdot2}\\~\\ =\,\frac{3^2\sqrt[5]{2}\,+\,2\sqrt3}{3\cdot2}\)
a = 3 and b = 2 so a + b = 3 + 2 = 5
+9466 \(\quad\,\frac{3}{\sqrt[5]{16}}\,+\,\frac{1}{\sqrt3} \\~\\ =\,\frac{3}{\sqrt[5]{2^4}}\,+\,\frac{1}{\sqrt3} \\~\\ =\,\frac{3}{2^\frac45}\,+\,\frac1{3^\frac12}\\~\\ =\,\frac{3\,\cdot\,2^\frac15}{2^\frac45\,\cdot\,2^\frac15}\,+\,\frac{1\,\cdot\,3^\frac12}{3^\frac12\,\cdot\,3^\frac12}\\~\\ =\,\frac{3\,\cdot\,2^\frac15}{2}\,+\,\frac{3^\frac12}{3}\\~\\ =\,\frac{3\,\cdot\,2^\frac15\,\cdot\,3}{2\,\cdot\,3}\,+\,\frac{3^\frac12\,\cdot\,2}{3\,\cdot\,2}\\~\\ =\,\frac{3^2\,\cdot\,2^\frac15}{3\cdot2}\,+\,\frac{3^\frac12\,\cdot\,2}{3\cdot2}\\~\\ =\,\frac{3^2\,\cdot\,2^\frac15\,+\,3^\frac12\,\cdot\,2}{3\cdot2}\\~\\ =\,\frac{3^2\sqrt[5]{2}\,+\,2\sqrt3}{3\cdot2}\)
a = 3 and b = 2 so a + b = 3 + 2 = 5
