Sure, let's simplify the expression [tex]\(\sqrt[4]{25 m^8 n^{16}}\)[/tex] step by step.
1. Identify the given expression:
[tex]\[\sqrt[4]{25 m^8 n^{16}}\][/tex]
2. Simplify each component inside the radical separately:
- Simplify 25:
[tex]\[25 = 5^2\][/tex]
Taking the fourth root:
[tex]\[\sqrt[4]{25} = \sqrt[4]{5^2} = \left(5^2\right)^{1/4} = 5^{\frac{2}{4}} = 5^{\frac{1}{2}} = \sqrt{5}\][/tex]
- Simplify [tex]\(m^8\)[/tex]:
[tex]\[m^8 = (m^8)^{1/4} = m^{8 \cdot \frac{1}{4}} = m^2\][/tex]
- Simplify [tex]\(n^{16}\)[/tex]:
[tex]\[n^{16} = (n^{16})^{1/4} = n^{16 \cdot \frac{1}{4}} = n^4\][/tex]
3. Combine the simplified parts:
[tex]\[\sqrt[4]{25} \cdot \sqrt[4]{m^8} \cdot \sqrt[4]{n^{16}} = \sqrt{5} \cdot m^2 \cdot n^4\][/tex]
Therefore, the simplified expression is:
[tex]\[m^2 n^4 \sqrt{5}\][/tex]
So, the correct simplified form of [tex]\(\sqrt[4]{25 m^8 n^{16}}\)[/tex] is:
[tex]\[m^2 n^4 \sqrt{5}\][/tex]
