To determine which expression is equivalent to [tex]\(\frac{(x^6 y^8)^3}{x^2 y^2}\)[/tex], let's break it down step by step.
1. Simplifying the Numerator:
[tex]\[
(x^6 y^8)^3
\][/tex]
Using the power of a power property [tex]\((a^m)^n = a^{mn}\)[/tex]:
[tex]\[
x^{6 \cdot 3} \cdot y^{8 \cdot 3} = x^{18} \cdot y^{24}
\][/tex]
2. Simplifying the Entire Fraction:
Substituting the simplified numerator back into the fraction:
[tex]\[
\frac{x^{18} y^{24}}{x^2 y^2}
\][/tex]
Using the division property of exponents [tex]\( \frac{a^m}{a^n} = a^{m-n} \)[/tex]:
[tex]\[
\frac{x^{18}}{x^2} \cdot \frac{y^{24}}{y^2} = x^{18-2} \cdot y^{24-2} = x^{16} \cdot y^{22}
\][/tex]
Thus, the equivalent expression is [tex]\(x^{16} y^{22}\)[/tex].
Therefore, the correct answer is:
[tex]\[
\boxed{x^{16} y^{22}}
\][/tex]
