sat-dog

Choice B is correct. For positive values of $a$ and $b$, $a^m{b}^m={\left(ab\right)}^m$, $\sqrt[n]{a}={\left(a\right)}^{\frac{1}{n}}$, and ${\left(a^j\right)}^k=a^{jk}$. Therefore, the given expression, $\sqrt[7]{x^9{y}^9}$, can be rewritten as $\sqrt[7]{{\left(xy\right)}^9}$. This expression is equivalent to ${\left({\left(xy\right)}^9\right)}^{\frac{1}{7}}$, which can be rewritten as ${\left(xy\right)}^{9·\frac{1}{7}}$, or ${\left(xy\right)}^{\frac{9}{7}}$. 

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.