the answer is much simpler than the other user might think
remember that [tex]\sqrt[n]{x^m}=x^\frac{m}{n}[/tex]
and [tex]\sqrt{ab}=(\sqrt{a})(\sqrt{b})[/tex]
so
[tex]\sqrt[3]{27x^{15}y^{24}}=[/tex]
[tex](\sqrt[3]{27})(\sqrt[3]{x^{15}})(\sqrt[3]{y^{24}})=[/tex]
factor them all. if they're factored already, then great
[tex](\sqrt[3]{3^3})(\sqrt[3]{x^{15}})(\sqrt[3]{y^{24}})=[/tex]
convert into fractional exponents
[tex](3^\frac{3}{3})(x^\frac{15}{3})(y^\frac{24}{3})=[/tex]
[tex](3^1)(x^5)(y^8)=[/tex]
[tex]3x^5y^8[/tex]
