Sure! Let's solve the expression step by step.
We need to find an equivalent expression for [tex]\(\sqrt{\frac{225}{625} m^4 n^6}\)[/tex].
1. Separate the constant and variable parts:
The given expression can be written as:
[tex]\[\sqrt{\frac{225}{625} \cdot m^4 \cdot n^6}\][/tex]
2. Simplify the constant part:
The fraction [tex]\(\frac{225}{625}\)[/tex] can be simplified:
[tex]\[\frac{225}{625} = \frac{15^2}{25^2} = \left(\frac{15}{25}\right)^2\][/tex]
[tex]\[\frac{15}{25} = \frac{3}{5}\][/tex]
Thus, we have:
[tex]\[\sqrt{\frac{225}{625}} = \sqrt{\left(\frac{3}{5}\right)^2} = \frac{3}{5}\][/tex]
3. Simplify the variable part:
- For [tex]\(m^4\)[/tex]:
[tex]\[\sqrt{m^4} = m^{4/2} = m^2\][/tex]
- For [tex]\(n^6\)[/tex]:
[tex]\[\sqrt{n^6} = n^{6/2} = n^3\][/tex]
4. Combine the simplified parts:
Now that we have simplified both the constant and variable parts, we combine them:
[tex]\[
\sqrt{\frac{225}{625} m^4 n^6} = \frac{3}{5} \cdot m^2 \cdot |n^3|
\][/tex]
5. Comparison with provided options:
- [tex]\(\frac{1}{20} m^2 \left|n^3\right|\)[/tex]
- [tex]\(\frac{1}{20} m^2 n^4\)[/tex]
- [tex]\(\frac{3}{5} m^2 \left|n^3\right|\)[/tex]
- [tex]\(\frac{3}{5} m^2 n^4\)[/tex]
The correct expression that matches our simplified result is:
[tex]\[
\frac{3}{5} m^2 |n^3|
\][/tex]
Thus, the equivalent expression is:
[tex]\(\boxed{\frac{3}{5} m^2\left|n^3\right|}\)[/tex]
