To determine which expression is equivalent to \(\frac{(x^6 y^8)^3}{x^2 y^2}\), let's go through the problem step by step:
1. Start with the given expression:
[tex]\[
\frac{(x^6 y^8)^3}{x^2 y^2}
\][/tex]
2. Simplify the numerator using the power rule \((a^m)^n = a^{m \cdot n}\):
[tex]\[
(x^6 y^8)^3 = x^{6 \cdot 3} y^{8 \cdot 3} = x^{18} y^{24}
\][/tex]
3. Rewrite the expression with the simplified numerator:
[tex]\[
\frac{x^{18} y^{24}}{x^2 y^2}
\][/tex]
4. Use the quotient rule for exponents, which states \(\frac{a^m}{a^n} = a^{m-n}\), to simplify the expression:
[tex]\[
\frac{x^{18}}{x^2} = x^{18-2} = x^{16}
\][/tex]
[tex]\[
\frac{y^{24}}{y^2} = y^{24-2} = y^{22}
\][/tex]
5. Combine the results:
[tex]\[
\frac{x^{18} y^{24}}{x^2 y^2} = x^{16} y^{22}
\][/tex]
Therefore, the equivalent expression is:
[tex]\(\boxed{x^{16} y^{22}}\)[/tex]
