To simplify the expression \(5 \frac{1}{2} \cdot 3\sqrt{125}\) using exponent properties, let's break it down step by step.
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Convert Mixed Number to Improper Fraction: \(5 \frac{1}{2} = 5 + \frac{1}{2} = \frac{10}{2} + \frac{1}{2} = \frac{11}{2}\).
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Simplify \(3\sqrt{125}\):
- First, simplify \(\sqrt{125}\): \[ \sqrt{125} = \sqrt{25 \cdot 5} = \sqrt{25} \cdot \sqrt{5} = 5\sqrt{5}. \]
- Now multiply by 3: \[ 3\sqrt{125} = 3 \cdot 5\sqrt{5} = 15\sqrt{5}. \]
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Multiply the Two Parts: Now multiply the two parts: \[ \frac{11}{2} \cdot 15\sqrt{5} = \frac{11 \cdot 15}{2} \cdot \sqrt{5} = \frac{165}{2}\sqrt{5}. \]
Next, let's analyze the given response options:
- Response 1: \(5\sqrt{5}\)
- Response 2: \(\frac{\sqrt{5}}{5}\)
- Response 3: \(\frac{1}{5}\)
- Response 4: \(505^0\)
None of these options directly correspond with \(\frac{165}{2}\sqrt{5}\), which would simplify to \(82.5\sqrt{5}\).
However, we need to look at each response and check for equivalent simplifications:
- \(5\sqrt{5} \neq \frac{165}{2}\sqrt{5}\)
- \(\frac{\sqrt{5}}{5} \neq \frac{165}{2}\sqrt{5}\)
- \(\frac{1}{5} \neq \frac{165}{2}\sqrt{5}\)
- \(505^0 = 1\) and does not equal \(\frac{165}{2}\sqrt{5}\).
Since none of these options appear to match an equivalent expression based on simplification, please double-check the original expression given for simplification or any overall conditions that may change the interpretation.
Based strictly on the simplifications we performed, none of the options provided are equivalent to the simplified version of the expression.