To find the cube root of [tex]\( 512 m^{12} n^{15} \)[/tex], let's break down each part of the expression.
1. Cube root of the coefficient [tex]\( 512 \)[/tex]:
[tex]\[
\sqrt[3]{512}
\][/tex]
We know that [tex]\( 512 \)[/tex] is [tex]\( 8^3 \)[/tex], so:
[tex]\[
\sqrt[3]{512} = 8
\][/tex]
2. Cube root of [tex]\( m^{12} \)[/tex]:
[tex]\[
\sqrt[3]{m^{12}} = m^{12/3} = m^4
\][/tex]
3. Cube root of [tex]\( n^{15} \)[/tex]:
[tex]\[
\sqrt[3]{n^{15}} = n^{15/3} = n^5
\][/tex]
Combining these results, we get:
[tex]\[
\sqrt[3]{512 m^{12} n^{15}} = 8 m^4 n^5
\][/tex]
So, the cube root of [tex]\( 512 m^{12} n^{15} \)[/tex] is:
[tex]\[
8 m^4 n^5
\][/tex]
Therefore, the correct choice from the given options is:
[tex]\[
\boxed{8 m^4 n^5}
\][/tex]
