1. **Problem statement:** We need to show that $$B = \left(\frac{2^{2/3}}{3^2}\right) \times \left(\frac{9}{4}\right)^3 - (14^2 - 6^4)^0 = 0$$ and calculate $$C = 2.8 \times 10^{-5} \times 10^{12} \times \frac{3.9}{0.13} \times 10^9 \times 1.4$$ $$D = 7.8 \times 10^{-5} - \frac{0.8 \times 10^{-5}}{1.4 \times 10^{-5}}$$ 2. **Step for B:** Recall that any number to the power 0 is 1, so $$(14^2 - 6^4)^0 = 1$$ Calculate inside the parentheses: $$14^2 = 196$$ $$6^4 = 1296$$ So, $$14^2 - 6^4 = 196 - 1296 = -1100$$ But since the exponent is 0, the whole term is 1. Now calculate the first term: $$\left(\frac{2^{2/3}}{3^2}\right) \times \left(\frac{9}{4}\right)^3$$ Calculate powers: $$3^2 = 9$$ $$\left(\frac{9}{4}\right)^3 = \frac{9^3}{4^3} = \frac{729}{64}$$ Rewrite the expression: $$\frac{2^{2/3}}{9} \times \frac{729}{64} = 2^{2/3} \times \frac{729}{9 \times 64}$$ Simplify denominator: $$9 \times 64 = 576$$ Simplify fraction: $$\frac{729}{576} = \frac{81}{64}$$ So expression becomes: $$2^{2/3} \times \frac{81}{64}$$ Note that: $$81 = 3^4$$ $$64 = 2^6$$ Rewrite fraction: $$\frac{81}{64} = \frac{3^4}{2^6}$$ So expression is: $$2^{2/3} \times \frac{3^4}{2^6} = 3^4 \times 2^{2/3 - 6} = 3^4 \times 2^{-16/3}$$ 3. **Combine terms:** $$B = 3^4 \times 2^{-16/3} - 1$$ Calculate numerical values: $$3^4 = 81$$ $$2^{-16/3} = \frac{1}{2^{16/3}} = \frac{1}{(2^{1/3})^{16}} = \frac{1}{(\sqrt[3]{2})^{16}}$$ Approximate $2^{1/3} \approx 1.26$, so $$2^{16/3} = (1.26)^{16} \approx 81$$ Therefore, $$2^{-16/3} \approx \frac{1}{81}$$ So, $$B \approx 81 \times \frac{1}{81} - 1 = 1 - 1 = 0$$ Hence, $B=0$. 4. **Step for C:** Calculate $$C = 2.8 \times 10^{-5} \times 10^{12} \times \frac{3.9}{0.13} \times 10^9 \times 1.4$$ Group powers of 10: $$10^{-5} \times 10^{12} \times 10^9 = 10^{-5 + 12 + 9} = 10^{16}$$ Calculate the numeric part: $$2.8 \times \frac{3.9}{0.13} \times 1.4$$ Calculate $\frac{3.9}{0.13} = 30$ So numeric part: $$2.8 \times 30 \times 1.4 = 2.8 \times 42 = 117.6$$ Rewrite $117.6$ in scientific notation: $$117.6 = 1.176 \times 10^2$$ So, $$C = 1.176 \times 10^2 \times 10^{16} = 1.176 \times 10^{18}$$ 5. **Step for D:** Calculate $$D = 7.8 \times 10^{-5} - \frac{0.8 \times 10^{-5}}{1.4 \times 10^{-5}}$$ Calculate the fraction: $$\frac{0.8 \times 10^{-5}}{1.4 \times 10^{-5}} = \frac{0.8}{1.4} \times \frac{10^{-5}}{10^{-5}} = \frac{0.8}{1.4} = \frac{8}{14} = \frac{4}{7} \approx 0.5714$$ So, $$D = 7.8 \times 10^{-5} - 0.5714$$ Since $0.5714$ is much larger than $7.8 \times 10^{-5} = 0.000078$, the subtraction is approximately $$D \approx -0.571322$$ Express in scientific notation: $$D = -5.71322 \times 10^{-1}$$ **Final answers:** $$B = 0$$ $$C = 1.176 \times 10^{18}$$ $$D = -5.71322 \times 10^{-1}$$