1. **State the problem:** Simplify the expression $$\left(\frac{16}{81}\right)^{-\frac{3}{4}} \times \sqrt{\frac{100}{81}}.$$\n\n2. **Recall the rules:**\n- For any positive number $a$ and rational exponent $m/n$, $$a^{m/n} = \sqrt[n]{a^m}.$$\n- Negative exponents mean reciprocal: $$a^{-x} = \frac{1}{a^x}.$$\n- Square root is the same as exponent $1/2$: $$\sqrt{a} = a^{1/2}.$$\n\n3. **Simplify each part:**\n- Simplify $$\left(\frac{16}{81}\right)^{-\frac{3}{4}} = \left(\frac{16}{81}\right)^{\frac{3}{4}}$$ reciprocal because of negative exponent.\n- Calculate $$\left(\frac{16}{81}\right)^{\frac{3}{4}} = \left(\left(\frac{16}{81}\right)^{\frac{1}{4}}\right)^3.$$\n- Find $$\left(\frac{16}{81}\right)^{\frac{1}{4}} = \frac{16^{1/4}}{81^{1/4}} = \frac{2}{3}$$ since $16 = 2^4$ and $81 = 3^4$.\n- Raise to the power 3: $$\left(\frac{2}{3}\right)^3 = \frac{8}{27}.$$\n- So, $$\left(\frac{16}{81}\right)^{-\frac{3}{4}} = \frac{27}{8}$$ (reciprocal of $\frac{8}{27}$).\n\n4. **Simplify the square root:**\n- $$\sqrt{\frac{100}{81}} = \frac{\sqrt{100}}{\sqrt{81}} = \frac{10}{9}.$$\n\n5. **Multiply the two results:**\n$$\frac{27}{8} \times \frac{10}{9} = \frac{27 \times 10}{8 \times 9} = \frac{270}{72}.$$\n\n6. **Simplify the fraction:**\n- Divide numerator and denominator by 18: $$\frac{270 \div 18}{72 \div 18} = \frac{15}{4}.$$\n\n**Final answer:** $$\boxed{\frac{15}{4}}.$$