Sure, let's simplify the following expression step-by-step:
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]
### Step 1: Group terms involving similar cube roots.
We can group the terms involving [tex]\(\sqrt[3]{2x}\)[/tex], [tex]\(\sqrt[3]{16x}\)[/tex], [tex]\(\sqrt[3]{8x}\)[/tex], and [tex]\(\sqrt[3]{x}\)[/tex].
### Step 2: Combine like terms.
#### Terms involving [tex]\(\sqrt[3]{2x}\)[/tex]:
[tex]\[ 7(\sqrt[3]{2x}) - 5(\sqrt[3]{2x}) + 1(\sqrt[3]{2x}) - 1(\sqrt[3]{2x}) \][/tex]
[tex]\[ (7 - 5 + 1 - 1)(\sqrt[3]{2x}) \][/tex]
[tex]\[ 2(\sqrt[3]{2x}) \][/tex]
#### Terms involving [tex]\(\sqrt[3]{8x}\)[/tex]:
[tex]\[ -3(\sqrt[3]{8x}) \][/tex]
Given that [tex]\(8 = 2^3\)[/tex], [tex]\(\sqrt[3]{8x} = (2^3 x)^{1/3} = 2 x^{1/3}\)[/tex].
Hence, it simplifies to:
[tex]\[ -3(2 x^{1/3}) = -6 x^{1/3} \][/tex]
#### Terms involving [tex]\(\sqrt[3]{16x}\)[/tex]:
[tex]\[ -3(\sqrt[3]{16x}) \][/tex]
Given that [tex]\(16 = 2^4\)[/tex], [tex]\(\sqrt[3]{16x} = (2^4 x)^{1/3} = 2^{4/3} x^{1/3} = (2^{1/3})^4 x^{1/3} = 2^{4/3} x^{1/3} \)[/tex].
So, this term simplifies to:
[tex]\[ -3(2^{4/3} x^{1/3}) \][/tex]
#### Terms involving [tex]\(\sqrt[3]{x}\)[/tex]:
[tex]\[ 5(\sqrt[3]{x}) - 6(\sqrt[3]{x}) - 8(\sqrt[3]{x}) \][/tex]
[tex]\[ (5 - 6 - 8)(\sqrt[3]{x}) \][/tex]
[tex]\[ -9(\sqrt[3]{x}) \][/tex]
### Step 3: Combine the simplified terms.
Summarizing all the like terms, we get:
[tex]\[ 2(\sqrt[3]{2x}) - 6 x^{1/3} - 3(2^{4/3} x^{1/3}) - 9(\sqrt[3]{x}) \][/tex]
Finally, evaluating the numerical components ([tex]\(\sqrt[3]{2}\)[/tex] and [tex]\(\sqrt[3]{16}\)[/tex]) as required:
[tex]\[ \{2 \cdot 2^{1/3}\} -6 - \{3 \cdot 2^{4/3}\} -9\][/tex]
### Final simplified form:
By evaluating these, we find the simplified numeric values:
[tex]\[ 2(\sqrt[3]{2}) - 6 - 3(\sqrt[3]{16}) - 9 = 2.5198420997897464 - 6 - 7.559526299369239 - 9 = -20.03968419957949 \][/tex]
So, the simplified form of the given expression is:
[tex]\[ -20.03968419957949 \][/tex]
