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 $72 = a^{3}.$

$$ \sqrt[3]{9000} = (9 \times 10^3)^{\frac{1}{3}} = 10 \times 9^{\frac{1}{3}}$$
$$ \sqrt[3]{576} = \sqrt[3]{(2^3 \times 3)^2} = 4 \times 9^{\frac{1}{3}}$$
$$ 6\sqrt[3]{9000} = 60 \times 9^{\frac{1}{3}}$$
$$ 7\sqrt[3]{576}  = 28 \times 9^{\frac{1}{3}} $$
$$ 7\sqrt[3]{576} +  6\sqrt[3]{9000}  = (60 \times 9^{\frac{1}{3}}) + (28 \times 9^{\frac{1}{3}})  = 88 \times 9^{\frac{1}{3}} $$
