Simplify ((16)/(81))^((1)/(4)) ÷ ((9)/(16))^(-(1)/(2))

Note that $16 = 2^{4}$ and $81 = 3^{4}$.

$$\frac{16}{81} = \frac{2^{4}}{3^{4}} = \left( \frac{2}{3} \right)^{4}$$ $$\left( \frac{16}{81} \right)^{\frac{1}{4}} = \left[ \left( \frac{2}{3} \right)^{4} \right]^{\frac{1}{4}} = \left( \frac{2}{3} \right)^{4 \cdot \frac{1}{4}} = \left( \frac{2}{3} \right)^{1} = \frac{2}{3}$$

Note that $9 = 3^{2}$ and $16 = 4^{2}$.

$$\frac{9}{16} = \frac{3^{2}}{4^{2}} = \left( \frac{3}{4} \right)^{2}$$ $$\left( \frac{9}{16} \right)^{\frac{1}{2}} = \left[ \left( \frac{3}{4} \right)^{2} \right]^{\frac{1}{2}} = \left( \frac{3}{4} \right)^{2 \cdot \frac{1}{2}} = \left( \frac{3}{4} \right)^{1} = \frac{3}{4}$$

$$\frac{2}{3} \cdot \frac{3}{4}$$

Substitute numerators and denominators:

$$\frac{2 \cdot 3}{3 \cdot 4}$$

Cancel common factor 3:

$$\frac{2}{4}$$

Simplify by dividing numerator and denominator by 2:

$$\frac{2 \div 2}{4 \div 2} = \frac{1}{2}$$

\frac{1}{2}