To simplify the given expression:
[tex]\[
7 \sqrt[3]{2x} - 3 \sqrt[3]{16x} - 3 \sqrt[3]{8x} - 5 \sqrt[3]{2x} + 5 \sqrt[3]{x} + \sqrt[3]{2x} - 6 \sqrt[3]{x} - \sqrt[3]{2x} - 8 \sqrt[3]{x}
\][/tex]
we can start by combining like terms.
First, let's identify the individual terms involving [tex]\(\sqrt[3]{2x}\)[/tex] and [tex]\(\sqrt[3]{x}\)[/tex].
1. Terms involving [tex]\(\sqrt[3]{2x}\)[/tex]:
[tex]\[
7 \sqrt[3]{2x} - 5 \sqrt[3]{2x} + \sqrt[3]{2x} - \sqrt[3]{2x}
\][/tex]
Combining these terms gives:
[tex]\[
(7 - 5 + 1 - 1) \sqrt[3]{2x} = 2 \sqrt[3]{2x}
\][/tex]
2. Terms involving [tex]\(\sqrt[3]{x}\)[/tex]:
[tex]\[
5 \sqrt[3]{x} - 6 \sqrt[3]{x} - 8 \sqrt[3]{x}
\][/tex]
Combining these terms gives:
[tex]\[
(5 - 6 - 8) \sqrt[3]{x} = -9 \sqrt[3]{x}
\][/tex]
3. For the remaining terms [tex]\(-3 \sqrt[3]{16x}\)[/tex] and [tex]\(-3 \sqrt[3]{8x}\)[/tex], let's break them down in terms of [tex]\(\sqrt[3]{2x}\)[/tex] and [tex]\(\sqrt[3]{x}\)[/tex]:
[tex]\[
-3 \sqrt[3]{16x} = -3 \cdot \sqrt[3]{2^4 \cdot x} = -3 \cdot 2^{4/3} \sqrt[3]{x}
\][/tex]
[tex]\[
-3 \sqrt[3]{8x} = -3 \cdot \sqrt[3]{2^3 \cdot x} = -3 \cdot 2 \sqrt[3]{2x} = -6 \sqrt[3]{2x}
\][/tex]
Now let's combine all terms:
[tex]\[
2 \sqrt[3]{2x} - 6 \sqrt[3]{2x} - 3 \cdot 2^{4/3} \sqrt[3]{x} -9 \sqrt[3]{x}
\][/tex]
Simplify the terms involving [tex]\(\sqrt[3]{2x}\)[/tex] and [tex]\(\sqrt[3]{x}\)[/tex]:
[tex]\[
(2 - 6) \sqrt[3]{2x} + (-9 - 3 \cdot 2^{4/3}) \sqrt[3]{x}
\][/tex]
This reduces to:
[tex]\[
-4 \sqrt[3]{2x} - (9 + 3 \cdot 2^{4/3}) \sqrt[3]{x}
\][/tex]
Factoring out [tex]\(\sqrt[3]{x}\)[/tex]:
[tex]\[
\sqrt[3]{x} \left( -4 \cdot 2^{1/3} - (9 + 3 \cdot 2^{4/3}) \right)
\][/tex]
Combining all constants and simplifying the entire expression by noticing that [tex]\(-4 \cdot 2^{1/3} - 9 - 3 \cdot 2^{4/3}\)[/tex] can be simplified to:
[tex]\[
\sqrt[3]{x}\left(-15 - 4 \cdot 2^{1/3}\right)
\][/tex]
Thus, the simplified form of the expression is:
[tex]\[
\boxed{x^{1/3}\left(-15 - 4 \cdot 2^{1/3}\right)}
\][/tex]
