Answer :
To simplify the given expression, we need to combine like terms. The terms involve different cube roots, some with [tex]\(\sqrt[3]{2x}\)[/tex] and others with [tex]\(\sqrt[3]{x}\)[/tex]. Here is the expression we need to simplify:
[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]
First, let's group the terms with the same cube roots together:
[tex]\[ (7(\sqrt[3]{2x}) - 5(\sqrt[3]{2x}) + \sqrt[3]{2x} - \sqrt[3]{2x}) + (-3(\sqrt[3]{16x})) + (-3(\sqrt[3]{8x})) + (5(\sqrt[3]{x}) - 6(\sqrt[3]{x}) - 8(\sqrt[3]{x})) \][/tex]
Now simplify within each group:
[tex]\[ (7 - 5 + 1 - 1) (\sqrt[3]{2x}) - 3(\sqrt[3]{16x}) - 3(\sqrt[3]{8x}) + (5 - 6 - 8)(\sqrt[3]{x}) \][/tex]
Simplify the coefficients for each term:
[tex]\[ 2 (\sqrt[3]{2x}) - 3 (\sqrt[3]{16x}) - 3 (\sqrt[3]{8x}) - 9 (\sqrt[3]{x}) \][/tex]
Next, simplify [tex]\(\sqrt[3]{16x}\)[/tex] and [tex]\(\sqrt[3]{8x}\)[/tex]:
[tex]\[ \sqrt[3]{16x} = \sqrt[3]{2^4 \cdot x} = 2^{4/3} \cdot \sqrt[3]{x} \][/tex]
[tex]\[ \sqrt[3]{8x} = \sqrt[3]{2^3 \cdot x} = 2 \cdot \sqrt[3]{x} \][/tex]
Substitute these simplified forms back into the expression:
[tex]\[ 2 (\sqrt[3]{2x}) - 3 (2^{4/3} (\sqrt[3]{x})) - 3 (2 (\sqrt[3]{x})) - 9 (\sqrt[3]{x}) \][/tex]
Distribute the coefficients:
[tex]\[ 2 (\sqrt[3]{2x}) - 3 \cdot 2^{4/3} (\sqrt[3]{x}) - 6 (\sqrt[3]{x}) - 9 (\sqrt[3]{x}) \][/tex]
Combine the terms involving [tex]\(\sqrt[3]{x}\)[/tex]:
[tex]\[ 2 (\sqrt[3]{2x}) - (3 \cdot 2^{4/3} \cdot \sqrt[3]{x} + 6 \cdot \sqrt[3]{x} + 9 \cdot \sqrt[3]{x}) \][/tex]
Since [tex]\(\sqrt[3]{2x}\)[/tex] is already combined, let's combine the coefficients of [tex]\(\sqrt[3]{x}\)[/tex]:
[tex]\[ 2 (\sqrt[3]{2x}) - \left( 3 \cdot 2^{4/3} + 6 + 9 \right) (\sqrt[3]{x}) \][/tex]
Adding the coefficients:
[tex]\[ 3 \cdot 2^{4/3} + 15 \][/tex]
Thus, the simplified form of the original expression is:
[tex]\[ x^{1/3}(-15 - 4 \cdot 2^{1/3}) \][/tex]
[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]
First, let's group the terms with the same cube roots together:
[tex]\[ (7(\sqrt[3]{2x}) - 5(\sqrt[3]{2x}) + \sqrt[3]{2x} - \sqrt[3]{2x}) + (-3(\sqrt[3]{16x})) + (-3(\sqrt[3]{8x})) + (5(\sqrt[3]{x}) - 6(\sqrt[3]{x}) - 8(\sqrt[3]{x})) \][/tex]
Now simplify within each group:
[tex]\[ (7 - 5 + 1 - 1) (\sqrt[3]{2x}) - 3(\sqrt[3]{16x}) - 3(\sqrt[3]{8x}) + (5 - 6 - 8)(\sqrt[3]{x}) \][/tex]
Simplify the coefficients for each term:
[tex]\[ 2 (\sqrt[3]{2x}) - 3 (\sqrt[3]{16x}) - 3 (\sqrt[3]{8x}) - 9 (\sqrt[3]{x}) \][/tex]
Next, simplify [tex]\(\sqrt[3]{16x}\)[/tex] and [tex]\(\sqrt[3]{8x}\)[/tex]:
[tex]\[ \sqrt[3]{16x} = \sqrt[3]{2^4 \cdot x} = 2^{4/3} \cdot \sqrt[3]{x} \][/tex]
[tex]\[ \sqrt[3]{8x} = \sqrt[3]{2^3 \cdot x} = 2 \cdot \sqrt[3]{x} \][/tex]
Substitute these simplified forms back into the expression:
[tex]\[ 2 (\sqrt[3]{2x}) - 3 (2^{4/3} (\sqrt[3]{x})) - 3 (2 (\sqrt[3]{x})) - 9 (\sqrt[3]{x}) \][/tex]
Distribute the coefficients:
[tex]\[ 2 (\sqrt[3]{2x}) - 3 \cdot 2^{4/3} (\sqrt[3]{x}) - 6 (\sqrt[3]{x}) - 9 (\sqrt[3]{x}) \][/tex]
Combine the terms involving [tex]\(\sqrt[3]{x}\)[/tex]:
[tex]\[ 2 (\sqrt[3]{2x}) - (3 \cdot 2^{4/3} \cdot \sqrt[3]{x} + 6 \cdot \sqrt[3]{x} + 9 \cdot \sqrt[3]{x}) \][/tex]
Since [tex]\(\sqrt[3]{2x}\)[/tex] is already combined, let's combine the coefficients of [tex]\(\sqrt[3]{x}\)[/tex]:
[tex]\[ 2 (\sqrt[3]{2x}) - \left( 3 \cdot 2^{4/3} + 6 + 9 \right) (\sqrt[3]{x}) \][/tex]
Adding the coefficients:
[tex]\[ 3 \cdot 2^{4/3} + 15 \][/tex]
Thus, the simplified form of the original expression is:
[tex]\[ x^{1/3}(-15 - 4 \cdot 2^{1/3}) \][/tex]
