To simplify the expression \(\frac{5^{\frac{1}{2}}}{\sqrt[3]{125}}\) using exponent properties, we first rewrite \(\sqrt[3]{125}\) in terms of exponents.
We know that \(125 = 5^3\). Therefore, we can express \(\sqrt[3]{125}\) as follows:
\[ \sqrt[3]{125} = \sqrt[3]{5^3} = 5^{3 \cdot \frac{1}{3}} = 5^1 = 5 \]
Now, we can substitute this back into the original expression:
\[ \frac{5^{\frac{1}{2}}}{\sqrt[3]{125}} = \frac{5^{\frac{1}{2}}}{5} \]
Using the properties of exponents, specifically \(\frac{a^m}{a^n} = a^{m-n}\), we can simplify this fraction:
\[ \frac{5^{\frac{1}{2}}}{5} = \frac{5^{\frac{1}{2}}}{5^1} = 5^{\frac{1}{2} - 1} = 5^{\frac{1}{2} - \frac{2}{2}} = 5^{-\frac{1}{2}} \]
Thus, the equivalent expression for \(\frac{5^{\frac{1}{2}}}{\sqrt[3]{125}}\) when simplified using exponent properties is:
\[ 5^{-\frac{1}{2}} \]
Alternatively, this can also be expressed as:
\[ \frac{1}{5^{\frac{1}{2}}} = \frac{1}{\sqrt{5}} \]
Both \(5^{-\frac{1}{2}}\) and \(\frac{1}{\sqrt{5}}\) are equivalent expressions to the original fraction.
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