Simplify nth Roots $6\sqrt[3]{9000} + 7\sqrt[3]{576}$ I'm having some concerns about how I am going about simplifying this radical expression. I wanted to know if this would be an accurate method of solving.
Simplify:
$$6\sqrt[3]{9000} + 7\sqrt[3]{576}$$
Radical expression product rule & greatest common factor:
$$6\sqrt[3]{125}\sqrt[3]{72} + 7\sqrt[3]{8}\sqrt[3]{72}$$
Principle of nth root:
$$6 \cdot 5\sqrt[3]{72} + 7 \cdot 2\sqrt[3]{72}$$
Simplify:
$$30\sqrt[3]{72} + 14\sqrt[3]{72}$$
Addition of radical rule:
$$30 + 14\sqrt[3]{72}$$
Simplify:
$$44\sqrt[3]{72}$$
Is $44\sqrt[3]{72}$ completely simplified?
I'm positive that 72 has no whole number $a$ that satisfies $72 = a^{3}.$

You write "I'm positive that 72 has no whole number $a$ that satisfies $72 = a^{3}$". However, simplifying also works if there $72$ has a cubic factor. And indeed, $72$ contains a cube $a^3=8$. So we get $88\cdot \sqrt[3]{9}$.
