To find the perimeter of a square, you can use the formula:
\[ P = 4 \times \text{side length} \]
Given the side length is \( 3\sqrt{32} \), we can calculate the perimeter as follows:
\[ P = 4 \times 3\sqrt{32} \]
This simplifies to:
\[ P = 12\sqrt{32} \]
Now, we can simplify \( \sqrt{32} \):
\[ \sqrt{32} = \sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2} = 4\sqrt{2} \]
So, substituting \( \sqrt{32} \) back into the perimeter equation:
\[ P = 12 \times 4\sqrt{2} = 48\sqrt{2} \]
Now, let's match this to the options provided. The perimeter in simplest radical form is \( 48\sqrt{2} \). However, it seems this specific option is not among your choices.
Given your options:
- \( 12\sqrt{128} \)
- \( 15\sqrt{3} \)
- \( 3\sqrt{32} \)
- \( 3\sqrt{2} \)
We should determine if any of these could be equivalent to \( 48\sqrt{2} \):
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For \( 12\sqrt{128} \): \[ \sqrt{128} = \sqrt{64 \times 2} = 8\sqrt{2} \quad \Rightarrow \quad 12\sqrt{128} = 12 \times 8\sqrt{2} = 96\sqrt{2} \]
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For \( 15\sqrt{3} \): No equivalent radical form to \( \sqrt{2} \).
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For \( 3\sqrt{32} \): \[ \sqrt{32} = 4\sqrt{2} \quad \Rightarrow \quad 3\sqrt{32} = 3 \times 4\sqrt{2} = 12\sqrt{2} \]
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For \( 3\sqrt{2} \): Simple radical with no equivalence to \( \sqrt{2} \).
Thus, \( 12\sqrt{32} = 48\sqrt{2} \) isn't actually among the options given, while \( 3\sqrt{32} = 12\sqrt{2} \).
If you want the perimeter in simplest radical, it is \( 48\sqrt{2} \), but based on the provided options, the closest correct calculation for \( 4 \times 3\sqrt{32} = 12\sqrt{32} \) would be \( 12\sqrt{128} \) if it equates \( 48\) as \( \sqrt{2} = 96\sqrt{2} \).
However, none matches the \( P = 48\sqrt{2} \). Please check the provided options once more. Would you like to proceed with these calculations again or need any further assistance?
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