What is the simplified form of the following expression?

[tex]\( 7(\sqrt[3]{2x}) - 3(\sqrt[3]{16x}) - 3(\sqrt[3]{8x}) \)[/tex]

A. [tex]\( -5(\sqrt[3]{2x}) \)[/tex]

B. [tex]\( 5(\sqrt[3]{x}) \)[/tex]

C. [tex]\( \sqrt[3]{2x} - 6(\sqrt[3]{x}) \)[/tex]

D. [tex]\( -(\sqrt[3]{2x}) - 8(\sqrt[3]{x}) \)[/tex]



Answer :

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]