Let's break down the given expressions step by step to put [tex]\( 3 \sqrt{32} - 2 \sqrt[3]{32} \)[/tex] into its simplest radical form and identify the correct equivalent expression from the given choices.
### Step-by-Step Simplification
1. Simplifying [tex]\( \sqrt{32} \)[/tex]:
[tex]\[
\sqrt{32} = \sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2} = 4 \sqrt{2}
\][/tex]
Therefore,
[tex]\[
3 \sqrt{32} = 3 \times 4 \sqrt{2} = 12 \sqrt{2}
\][/tex]
2. Simplifying [tex]\( \sqrt[3]{32} \)[/tex]:
[tex]\[
\sqrt[3]{32} = \sqrt[3]{8 \times 4} = \sqrt[3]{8} \times \sqrt[3]{4} = 2 \sqrt[3]{4}
\][/tex]
Therefore,
[tex]\[
2 \sqrt[3]{32} = 2 \times 2 \sqrt[3]{4} = 4 \sqrt[3]{4}
\][/tex]
3. Combining the Simplified Expressions:
Now we combine these results:
[tex]\[
3 \sqrt{32} - 2 \sqrt[3]{32} = 12 \sqrt{2} - 4 \sqrt[3]{4}
\][/tex]
### Identifying the Correct Choice
We now compare our result [tex]\( 12 \sqrt{2} - 4 \sqrt[3]{4} \)[/tex] with the given choices:
1. [tex]\( 12 \sqrt{2} - 4 \sqrt[3]{4} \)[/tex]
2. [tex]\( 12 \sqrt{2} - 4 \sqrt[3]{16} \)[/tex]
3. [tex]\( 12 \sqrt{8} - 4 \sqrt[3]{16} \)[/tex]
4. [tex]\( 12 \sqrt{8} - 4 \sqrt[3]{4} \)[/tex]
From our simplification, we can see that the expression in the simplest radical form matches directly with the first choice:
[tex]\[
12 \sqrt{2} - 4 \sqrt[3]{4}
\][/tex]
### Conclusion
Thus, the simplest radical form of [tex]\( 3 \sqrt{32} - 2 \sqrt[3]{32} \)[/tex] is:
[tex]\[
12 \sqrt{2} - 4 \sqrt[3]{4}
\][/tex]
And this corresponds to the first choice listed.
