To solve this problem, we need to calculate and compare the given expressions step-by-step. Let's go through each part of the question in detail.
1. Compute the quotient:
[tex]\[
\frac{6 - 3(\sqrt[3]{6})}{\sqrt[3]{9}}
\][/tex]
Numerator: [tex]\( 6 - 3(\sqrt[3]{6}) \)[/tex]
Denominator: [tex]\( \sqrt[3]{9} \)[/tex]
Numerical result: [tex]\( 0.26375774640592026 \)[/tex]
2. Calculate each of the choices:
- Choice 1: [tex]\( 2(\sqrt[3]{3}) - \sqrt[3]{18} \)[/tex]
Numerical result: [tex]\( 0.2637577464059202 \)[/tex]
- Choice 2: [tex]\( 2(\sqrt[3]{3}) - 3(\sqrt[3]{2}) \)[/tex]
Numerical result: [tex]\( -0.8952640090698027 \)[/tex]
- Choice 3: [tex]\( 3(\sqrt[3]{3}) - \sqrt[3]{18} \)[/tex]
Numerical result: [tex]\( 1.706007316713328 \)[/tex]
- Choice 4: [tex]\( 3(\sqrt{3}) - 3(\sqrt{2}) \)[/tex]
Numerical result: [tex]\( 0.9535117355873464 \)[/tex]
From these calculations, we can see that the quotient [tex]\( \frac{6 - 3(\sqrt[3]{6})}{\sqrt[3]{9}} \)[/tex] (which is approximately [tex]\( 0.26375774640592026 \)[/tex]) closely matches the value of Choice 1: [tex]\( 2(\sqrt[3]{3}) - \sqrt[3]{18} \)[/tex] (which is approximately [tex]\( 0.2637577464059202 \)[/tex]).
Therefore, the correct answer is:
[tex]\[
2(\sqrt[3]{3}) - \sqrt[3]{18}
\][/tex]
