化简 $y^2 \div 9x^{-2} \times x^7$。 化简 $(81a^{12})^{5/4}$。

1. Rewrite the division as multiplication by the reciprocal.

$y^2 \div 9x^{-2} \times x^7 = y^2 \times \dfrac{1}{9x^{-2}} \times x^7$

2. Simplify the negative index and combine powers of $x$.

$\dfrac{1}{9x^{-2}} = \dfrac{x^2}{9}$

So,
$y^2 \times \dfrac{x^2}{9} \times x^7 = \dfrac{x^{2+7}y^2}{9} = \dfrac{x^9y^2}{9}$

1. Apply the power to each factor and simplify the indices.

$(81a^{12})^{\frac54}=81^{\frac54}\times (a^{12})^{\frac54}$
Since $81=3^4$,
$81^{\frac54}=(3^4)^{\frac54}=3^5=243$
Also,
$(a^{12})^{\frac54}=a^{12\times \frac54}=a^{15}$
Therefore,
$(81a^{12})^{\frac54}=243a^{15}.$