The problem asks us to simplify the expression $3\sqrt[3]{72x^5} - 2\sqrt[3]{18x} - 9\sqrt[3]{512x^3}$ and state any restrictions on the variable $x$.

1. Problem Description

The problem asks us to simplify the expression $3\sqrt[3]{72x^5} - 2\sqrt[3]{18x} - 9\sqrt[3]{512x^3}$ and state any restrictions on the variable $x$.

2. Solution Steps

First, we simplify each term separately:

$3\sqrt[3]{72x^5} = 3\sqrt[3]{8 \cdot 9 \cdot x^3 \cdot x^2} = 3 \cdot 2 \cdot x \cdot \sqrt[3]{9x^2} = 6x\sqrt[3]{9x^2}$

$2\sqrt[3]{18x} = 2\sqrt[3]{18x}$

$9\sqrt[3]{512x^3} = 9 \cdot \sqrt[3]{512} \cdot \sqrt[3]{x^3} = 9 \cdot 8 \cdot x = 72x$

Now, we rewrite the expression as:

$6x\sqrt[3]{9x^2} - 2\sqrt[3]{18x} - 72x$

Notice that we cannot combine the terms directly since the radicals are different. However, let's reconsider the first two terms:

$3\sqrt[3]{72x^5} = 3\sqrt[3]{8 \cdot 9x^3 x^2} = 3 \cdot 2x \sqrt[3]{9x^2} = 6x\sqrt[3]{9x^2}$

$2\sqrt[3]{18x} = 2\sqrt[3]{2 \cdot 9x}$

$9\sqrt[3]{512x^3} = 9(8x) = 72x$

So the original expression is $6x\sqrt[3]{9x^2} - 2\sqrt[3]{18x} - 72x$.

Since we are taking the cube root, there are no restrictions on the variable $x$. $x$ can be any real number.

There appears to be no further simplification possible.

Let's double-check the given problem. The original problem is:

$3\sqrt[3]{72x^5} - 2\sqrt[3]{18x} - 9\sqrt[3]{512x^3}$

$3 \sqrt[3]{2^3 \cdot 9 x^3 x^2} - 2\sqrt[3]{18x} - 9 \sqrt[3]{8^3 x^3}$

$3(2x)\sqrt[3]{9x^2} - 2\sqrt[3]{18x} - 9(8x)$

$6x\sqrt[3]{9x^2} - 2\sqrt[3]{18x} - 72x$

3. Final Answer

$6x\sqrt[3]{9x^2} - 2\sqrt[3]{18x} - 72x$

Restriction: No restriction on $x$. $x$ can be any real number.