1. **State the problem:** Find the value of the expression $$\left(1 - \frac{2}{3}\right)^{10} \cdot (0.6)^8$$. 2. **Simplify inside the parentheses:** $$1 - \frac{2}{3} = \frac{3}{3} - \frac{2}{3} = \frac{3 - 2}{3} = \frac{1}{3}$$ 3. **Rewrite the expression:** $$\left(\frac{1}{3}\right)^{10} \cdot (0.6)^8$$ 4. **Convert decimal to fraction for clarity:** $$0.6 = \frac{6}{10} = \frac{3}{5}$$ 5. **Rewrite the expression with fractions:** $$\left(\frac{1}{3}\right)^{10} \cdot \left(\frac{3}{5}\right)^8$$ 6. **Calculate powers separately:** $$\left(\frac{1}{3}\right)^{10} = \frac{1^{10}}{3^{10}} = \frac{1}{3^{10}}$$ $$\left(\frac{3}{5}\right)^8 = \frac{3^8}{5^8}$$ 7. **Multiply the two fractions:** $$\frac{1}{3^{10}} \cdot \frac{3^8}{5^8} = \frac{1 \cdot 3^8}{3^{10} \cdot 5^8} = \frac{3^8}{3^{10} \cdot 5^8}$$ 8. **Simplify the powers of 3:** $$\frac{3^8}{3^{10}} = \frac{\cancel{3^8} \cdot 1}{\cancel{3^8} \cdot 3^2} = \frac{1}{3^2} = \frac{1}{9}$$ 9. **Final expression:** $$\frac{1}{9 \cdot 5^8} = \frac{1}{9 \cdot 390625} = \frac{1}{3515625}$$ 10. **Decimal approximation:** $$\frac{1}{3515625} \approx 2.846 \times 10^{-7}$$ **Answer:** $$\boxed{\frac{1}{3515625} \approx 2.846 \times 10^{-7}}$$