To find which expression is equivalent to [tex]\(\frac{\left(x^6 y^8\right)^3}{x^2 y^2}\)[/tex], let's proceed step by step.
First, we need to simplify the numerator [tex]\(\left(x^6 y^8\right)^3\)[/tex]:
1. Exponentiation of each term inside the parentheses:
[tex]\[
\left(x^6 y^8\right)^3
\][/tex]
This can be simplified by raising each individual factor to the power of 3. Using the laws of exponents [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
[tex]\[
(x^6)^3 \cdot (y^8)^3 = x^{6 \cdot 3} \cdot y^{8 \cdot 3} = x^{18} \cdot y^{24}
\][/tex]
So, the numerator becomes:
[tex]\[
x^{18} y^{24}
\][/tex]
2. Simplifying the expression [tex]\(\frac{x^{18} y^{24}}{x^2 y^2}\)[/tex]:
We need to divide each term in the numerator by the corresponding term in the denominator. Using the property of exponents [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex]:
[tex]\[
\frac{x^{18}}{x^2} \cdot \frac{y^{24}}{y^2}
\][/tex]
Applying the property to each term:
[tex]\[
x^{18-2} \cdot y^{24-2} = x^{16} \cdot y^{22}
\][/tex]
Therefore, the simplified expression is:
[tex]\[
x^{16} y^{22}
\][/tex]
Thus, the expression [tex]\(\frac{\left(x^6 y^8\right)^3}{x^2 y^2}\)[/tex] is equivalent to [tex]\(x^{16} y^{22}\)[/tex].
The correct answer is:
[tex]\[
x^{16} y^{22}
\][/tex]
