To convert each of the given expressions into radical form, follow these steps.
### First Expression:
[tex]\[
(2 m^2 n)^{\frac{3}{2}}
\][/tex]
In radical form, an exponent of [tex]\(\frac{3}{2}\)[/tex] indicates a square root (since the denominator is 2) and then cubing the result (since the numerator is 3). Therefore:
[tex]\[
(2 m^2 n)^{\frac{3}{2}} = \sqrt{(2 m^2 n)^3} = \sqrt{2^3 m^6 n^3} = \sqrt{8 m^6 n^3}
\][/tex]
### Second Expression:
[tex]\[
\sqrt[3]{4 m^4 n^2}
\][/tex]
In radical form, [tex]\(\sqrt[3]{4 m^4 n^2}\)[/tex] represents the cube root. Therefore:
[tex]\[
\sqrt[3]{4 m^4 n^2} = (4 m^4 n^2)^{\frac{1}{3}}
\][/tex]
To simplify further:
[tex]\[
\sqrt[3]{4 m^4 n^2} = \sqrt[3]{4 \cdot m^4 \cdot n^2} = \sqrt[3]{4} \cdot \sqrt[3]{m^4} \cdot \sqrt[3]{n^2}
= \sqrt[3]{4} \cdot m^{4/3} \cdot n^{2/3}
\][/tex]
### Third Expression:
[tex]\[
\sqrt{8 m^6 n^3}
\][/tex]
In radical form, this expression is already simplified and remains:
[tex]\( \sqrt{8 m^6 n^3} \)[/tex]
### Fourth Expression:
[tex]\[
\sqrt{2 m^6 n^3}
\][/tex]
Similar to the third expression, this is already in radical form and remains:
[tex]\( \sqrt{2 m^6 n^3} \)[/tex]
### Fifth Expression:
[tex]\[
\sqrt[3]{4 m^2 n^2}
\][/tex]
In radical form, [tex]\(\sqrt[3]{4 m^2 n^2}\)[/tex] represents the cube root. Therefore:
[tex]\[
\sqrt[3]{4 m^2 n^2} = (4 m^2 n^2)^{\frac{1}{3}}
\][/tex]
To simplify further,
[tex]\[
\sqrt[3]{4 m^2 n^2} = \sqrt[3]{4 \cdot m^2 \cdot n^2} = \sqrt[3]{4} \cdot \sqrt[3]{m^2} \cdot \sqrt[3]{n^2}
= \sqrt[3]{4} \cdot m^{2/3} \cdot n^{2/3}
\][/tex]
### Conclusion:
After finding the radical forms for each expression, we have:
1. [tex]\((2 m^2 n)^{\frac{3}{2}} = \sqrt{8 m^6 n^3}\)[/tex]
2. [tex]\(\sqrt[3]{4 m^4 n^2} = \sqrt[3]{4} \cdot m^{4/3} \cdot n^{2/3}\)[/tex]
3. [tex]\(\sqrt{8 m^6 n^3}\)[/tex]
4. [tex]\(\sqrt{2 m^6 n^3}\)[/tex]
5. [tex]\(\sqrt[3]{4 m^2 n^2} = \sqrt[3]{4} \cdot m^{2/3} \cdot n^{2/3}\)[/tex]
Thus, the expressions in their radical forms are:
1. [tex]\(\sqrt{8 m^6 n^3}\)[/tex]
2. [tex]\(\sqrt[3]{4 m^4 n^2}\)[/tex]
3. [tex]\(\sqrt{8 m^6 n^3}\)[/tex]
4. [tex]\(\sqrt{2 m^6 n^3}\)[/tex]
5. [tex]\(\sqrt[3]{4 m^2 n^2}\)[/tex]
