To simplify the given expression, we need to apply the principles of simplifying cube roots and algebraic expressions step-by-step. The given expression is:
[tex]\[
\sqrt[3]{24 x^6 y^2} - 2 x^2 \sqrt[3]{375 y^2} - 3 x \sqrt[3]{16 x^3 y^2}
\][/tex]
Let's handle each term one by one.
### Term 1: [tex]\(\sqrt[3]{24 x^6 y^2}\)[/tex]
We can break this term down as follows:
[tex]\[
\sqrt[3]{24 x^6 y^2} = \sqrt[3]{24} \cdot \sqrt[3]{x^6} \cdot \sqrt[3]{y^2}
\][/tex]
Simplify each part:
- [tex]\(\sqrt[3]{24}\)[/tex] is a cube root that can be approximated, but let's leave it as is.
- [tex]\(\sqrt[3]{x^6} = x^{6/3} = x^2\)[/tex]
- [tex]\(\sqrt[3]{y^2}\)[/tex] is a part we can leave under the cube root.
So the term becomes:
[tex]\[
\sqrt[3]{24} \cdot x^2 \cdot \sqrt[3]{y^2}
\][/tex]
### Term 2: [tex]\(2 x^2 \sqrt[3]{375 y^2}\)[/tex]
Similarly, we break this term down:
[tex]\[
2 x^2 \sqrt[3]{375 y^2} = 2 x^2 \cdot \sqrt[3]{375} \cdot \sqrt[3]{y^2}
\][/tex]
Simplify each part:
- [tex]\(2 x^2\)[/tex] remains as is.
- [tex]\(\sqrt[3]{375}\)[/tex] is a cube root that can be approximated, but let's leave it as is.
- [tex]\(\sqrt[3]{y^2}\)[/tex] is a part we can leave under the cube root.
So the term becomes:
[tex]\[
2 x^2 \cdot \sqrt[3]{375} \cdot \sqrt[3]{y^2}
\][/tex]
### Term 3: [tex]\(3 x \sqrt[3]{16 x^3 y^2}\)[/tex]
Break it down:
[tex]\[
3 x \sqrt[3]{16 x^3 y^2} = 3 x \cdot \sqrt[3]{16} \cdot \sqrt[3]{x^3} \cdot \sqrt[3]{y^2}
\][/tex]
Simplify each part:
- [tex]\(3 x\)[/tex] remains as is.
- [tex]\(\sqrt[3]{16}\)[/tex] is a cube root that can be approximated, but let's leave it as is.
- [tex]\(\sqrt[3]{x^3} = x\)[/tex]
- [tex]\(\sqrt[3]{y^2}\)[/tex] is a part we can leave under the cube root.
So the term becomes:
[tex]\[
3 x \cdot \sqrt[3]{16} \cdot x \cdot \sqrt[3]{y^2} = 3 x^2 \cdot \sqrt[3]{16} \cdot \sqrt[3]{y^2}
\][/tex]
### Combined Expression
Now, combining everything back together, we have:
[tex]\[
\sqrt[3]{24} \cdot x^2 \cdot \sqrt[3]{y^2} - 2 x^2 \cdot \sqrt[3]{375} \cdot \sqrt[3]{y^2} - 3 x^2 \cdot \sqrt[3]{16} \cdot \sqrt[3]{y^2}
\][/tex]
Notice that each term has the common factor [tex]\(x^2 \cdot \sqrt[3]{y^2}\)[/tex]:
[tex]\[
x^2 \cdot \sqrt[3]{y^2} \left( \sqrt[3]{24} - 2 \sqrt[3]{375} - 3 \sqrt[3]{16} \right)
\][/tex]
This is the simplified form of the given expression:
[tex]\[
\boxed{x^2 \cdot \sqrt[3]{y^2} \left( \sqrt[3]{24} - 2 \sqrt[3]{375} - 3 \sqrt[3]{16} \right)}
\][/tex]
