25. $$ \sqrt[\ell]{\frac{9}{25}} $$

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UpStudy AI · Accepted Answer

To simplify the expression $$ \sqrt[\ell]{\frac{9}{25}} $$, follow these steps:

1. **Express the Fraction in Terms of Exponents:**

The fraction $$\frac{9}{25}$$ can be rewritten using exponents:
   $$
   \frac{9}{25} = \frac{3^2}{5^2}
   $$
   
2. **Apply the $$\ell$$th Root to the Fraction:**

The $$\ell$$th root of a fraction is equal to the fraction of the $$\ell$$th roots of the numerator and the denominator:
   $$
   \sqrt[\ell]{\frac{3^2}{5^2}} = \frac{\sqrt[\ell]{3^2}}{\sqrt[\ell]{5^2}}
   $$
   
3. **Simplify Using Exponent Rules:**

Recall that $$\sqrt[\ell]{a^b} = a^{b/\ell}$$. Applying this rule:
   $$
   \frac{\sqrt[\ell]{3^2}}{\sqrt[\ell]{5^2}} = \frac{3^{2/\ell}}{5^{2/\ell}}
   $$
   
4. **Express as a Single Exponent:**

This can also be written by combining the exponents:
   $$
   \frac{3^{2/\ell}}{5^{2/\ell}} = \left(\frac{3}{5}\right)^{2/\ell}
   $$
   
5. **Final Simplified Form:**

Therefore, the simplified form of the original expression is:
   $$
   \sqrt[\ell]{\frac{9}{25}} = \left(\frac{3}{5}\right)^{\frac{2}{\ell}}
   $$

**Summary:**
$$
\sqrt[\ell]{\frac{9}{25}} = \left(\frac{3}{5}\right)^{\frac{2}{\ell}}
$$
$$ \sqrt[\ell]{\frac{9}{25}} = \left(\frac{3}{5}\right)^{\frac{2}{\ell}} $$

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