1. **State the problem:** Evaluate the expression $$5 \frac{27}{3} \times \left(\frac{81}{16}\right)^{-\frac{1}{4}}$$ and simplify the expression $$\frac{1}{6.43 \times 6.43} + \frac{2}{3.56} - \frac{2}{3.56} + \frac{1}{8.51} - \frac{1}{8.51}$$. 2. **Rewrite mixed numbers and simplify:** - Convert $$5 \frac{27}{3}$$ to an improper fraction. Since $$\frac{27}{3} = 9$$, the expression becomes $$5 + 9 = 14$$. 3. **Simplify the power:** - Calculate $$\left(\frac{81}{16}\right)^{-\frac{1}{4}}$$. - Recall that $$a^{-b} = \frac{1}{a^b}$$, so $$\left(\frac{81}{16}\right)^{-\frac{1}{4}} = \frac{1}{\left(\frac{81}{16}\right)^{\frac{1}{4}}}$$. 4. **Calculate the fourth root:** - $$\left(\frac{81}{16}\right)^{\frac{1}{4}} = \frac{81^{\frac{1}{4}}}{16^{\frac{1}{4}}}$$. - Since $$81 = 3^4$$, $$81^{\frac{1}{4}} = 3$$. - Since $$16 = 2^4$$, $$16^{\frac{1}{4}} = 2$$. - Therefore, $$\left(\frac{81}{16}\right)^{\frac{1}{4}} = \frac{3}{2}$$. 5. **Evaluate the negative power:** - $$\left(\frac{81}{16}\right)^{-\frac{1}{4}} = \frac{1}{\frac{3}{2}} = \frac{2}{3}$$. 6. **Multiply:** - $$14 \times \frac{2}{3} = \frac{28}{3} = 9 \frac{1}{3}$$. 7. **Simplify the second expression:** - $$\frac{1}{6.43 \times 6.43} + \frac{2}{3.56} - \frac{2}{3.56} + \frac{1}{8.51} - \frac{1}{8.51}$$. - Notice that $$\frac{2}{3.56} - \frac{2}{3.56} = 0$$ and $$\frac{1}{8.51} - \frac{1}{8.51} = 0$$. - So the expression reduces to $$\frac{1}{6.43 \times 6.43}$$. 8. **Calculate:** - $$6.43 \times 6.43 = 41.3449$$. - Therefore, $$\frac{1}{41.3449} \approx 0.0242$$. **Final answers:** - $$5 \frac{27}{3} \times \left(\frac{81}{16}\right)^{-\frac{1}{4}} = 9 \frac{1}{3}$$. - $$\frac{1}{6.43 \times 6.43} + \frac{2}{3.56} - \frac{2}{3.56} + \frac{1}{8.51} - \frac{1}{8.51} \approx 0.0242$$.