Alright, let's simplify the given expression step-by-step:
[tex]\[ \sqrt[3]{36 x^2 \sqrt{12 x \sqrt{9 x^2}}} \][/tex]
1. Simplify the innermost part: Start by simplifying the innermost square root.
[tex]\[ \sqrt{9 x^2} \][/tex]
The square root of [tex]\(9 x^2\)[/tex] is:
[tex]\[ \sqrt{9 x^2} = 3x \][/tex]
2. Substitute and simplify further: Now, substitute [tex]\(3x\)[/tex] back into the expression.
[tex]\[ \sqrt[3]{36 x^2 \sqrt{12 x (3x)}} \][/tex]
And then multiply inside the square root:
[tex]\[ 12 x (3x) = 36 x^2 \][/tex]
So, the expression becomes:
[tex]\[ \sqrt[3]{36 x^2 \sqrt{36 x^2}} \][/tex]
3. Simplify the square root:
[tex]\[ \sqrt{36 x^2} \][/tex]
The square root of [tex]\(36 x^2\)[/tex] is:
[tex]\[ \sqrt{36 x^2} = 6x \][/tex]
Now substitute [tex]\(6x\)[/tex] back in:
[tex]\[ \sqrt[3]{36 x^2 (6x)} \][/tex]
4. Multiply the terms inside the cube root:
[tex]\[ 36 x^2 (6x) = 216 x^3 \][/tex]
So now we have:
[tex]\[ \sqrt[3]{216 x^3} \][/tex]
5. Simplify the cube root:
The cube root of [tex]\(216 x^3\)[/tex] is:
[tex]\[ \sqrt[3]{216 x^3} = 6x \][/tex]
Therefore, the simplified form of the given expression is:
[tex]\[ 6 x \][/tex]
Hence, the original expression simplifies to:
[tex]\[ \boxed{6 x} \][/tex]
