Let's simplify the expression [tex]\(\sqrt[4]{100 m^{12} n^4}\)[/tex] step-by-step.
1. Recognize the structure inside the root: The given expression is [tex]\(\sqrt[4]{100 m^{12} n^4}\)[/tex].
2. Break down each part under the root:
- [tex]\(100\)[/tex] is just a number.
- [tex]\(m^{12}\)[/tex] is the variable [tex]\(m\)[/tex] raised to the power of 12.
- [tex]\(n^4\)[/tex] is the variable [tex]\(n\)[/tex] raised to the power of 4.
3. Rewrite [tex]\(m^{12}\)[/tex] and [tex]\(n^4\)[/tex] in a more convenient form for taking the fourth root:
- [tex]\(m^{12}\)[/tex] can be rewritten as [tex]\((m^3)^4\)[/tex].
- [tex]\(n^4\)[/tex] is already in the form of a fourth power, i.e., [tex]\((n)^4\)[/tex].
4. Use the properties of roots: The fourth root of a product is the product of the fourth roots.
- The fourth root of [tex]\(100\)[/tex] is [tex]\(\sqrt[4]{100}\)[/tex].
- The fourth root of [tex]\((m^3)^4\)[/tex] is [tex]\(m^3\)[/tex] (since [tex]\(\sqrt[4]{(m^3)^4} = m^3\)[/tex]).
- The fourth root of [tex]\(n^4\)[/tex] is [tex]\(n\)[/tex] (since [tex]\(\sqrt[4]{(n^4)} = n\)[/tex]).
5. Simplify [tex]\(\sqrt[4]{100}\)[/tex]:
- [tex]\(100 = 10^2\)[/tex], so [tex]\(\sqrt[4]{100} = \sqrt[4]{10^2} = (10^2)^{1/4} = 10^{1/2} = \sqrt{10}\)[/tex].
6. Combine all parts together:
- [tex]\(\sqrt[4]{100} = \sqrt{10}\)[/tex],
- [tex]\(\sqrt[4]{(m^3)^4} = m^3\)[/tex],
- [tex]\(\sqrt[4]{n^4} = n\)[/tex].
Therefore,
[tex]\[
\sqrt[4]{100 m^{12} n^4} = \sqrt{10} \cdot m^3 \cdot n.
\][/tex]
So, the simplified expression is [tex]\(m^3 n \sqrt{10}\)[/tex].
Therefore, the final answer to the simplification problem [tex]\(\sqrt[4]{100 m^{12} n^4}\)[/tex] given [tex]\(m \geq 0\)[/tex] and [tex]\(n \geq 0\)[/tex] is:
[tex]\[
m^3 n \sqrt{10}.
\][/tex]
