To find an equivalent expression for \(\frac{5^{1/2}}{3\sqrt{125}}\), we will simplify the expression step by step using exponent properties and radical rules.
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Simplify \(\sqrt{125}\) first: \[ \sqrt{125} = \sqrt{25 \cdot 5} = \sqrt{25} \cdot \sqrt{5} = 5\sqrt{5} \]
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Now substitute \(\sqrt{125}\) back into the expression: \[ \frac{5^{1/2}}{3\sqrt{125}} = \frac{5^{1/2}}{3(5\sqrt{5})} \]
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Combine the terms in the denominator: \[ = \frac{5^{1/2}}{15\sqrt{5}} \]
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Rewrite \(\sqrt{5}\) as \(5^{1/2}\): \[ = \frac{5^{1/2}}{15 \cdot 5^{1/2}} \]
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Now, cancel \(5^{1/2}\) in the numerator and denominator: \[ = \frac{1}{15} \]
Thus, the simplified expression is \(\frac{1}{15}\).
Now, let’s compare this with the given options:
- Square root 5 over 5: \(\frac{\sqrt{5}}{5}\)
- 1: \(1\)
- 5 with power of zero: \(5^0 = 1\)
Since we found the simplified expression to be \(\frac{1}{15}\), none of the options given (\(\sqrt{5}/5\), \(1\), \(5^0\)) are equivalent to our simplified result.
Hence, none of the provided options are correct. If there are additional options you wish to check, please provide them.