(3) \( \frac{\left(2 \times 10^{2}\right)^{3}\left(6 \times 10^{-2}\right)^{2}}{\left(2 \times 10^{-2}\right)\left(3 \times 10^{4}\right)^{2}}= \)

Calculate the value by following steps: - step0: Calculate: \(\frac{\left(2\times 10^{2}\right)^{3}\left(6\times 10^{-2}\right)^{2}}{\left(\left(2\times 10^{-2}\right)\left(3\times 10^{4}\right)^{2}\right)}\) - step1: Evaluate: \(\frac{\left(2\times 10^{2}\right)^{3}\left(6\times 10^{-2}\right)^{2}}{\left(2\times 10^{-2}\left(3\times 10^{4}\right)^{2}\right)}\) - step2: Remove the parentheses: \(\frac{\left(2\times 10^{2}\right)^{3}\left(6\times 10^{-2}\right)^{2}}{2\times 10^{-2}\left(3\times 10^{4}\right)^{2}}\) - step3: Multiply the numbers: \(\frac{\frac{9}{2500}\left(2\times 10^{2}\right)^{3}}{2\times 10^{-2}\left(3\times 10^{4}\right)^{2}}\) - step4: Multiply the numbers: \(\frac{\frac{9}{2500}\left(2\times 10^{2}\right)^{3}}{\frac{\left(3\times 10^{4}\right)^{2}}{50}}\) - step5: Evaluate the power: \(\frac{28800}{\frac{\left(3\times 10^{4}\right)^{2}}{50}}\) - step6: Evaluate the power: \(\frac{28800}{18000000}\) - step7: Multiply by the reciprocal: \(28800\times \frac{1}{18000000}\) - step8: Reduce the numbers: \(1\times \frac{1}{625}\) - step9: Multiply: \(\frac{1}{625}\) Para resolver la expresión \[ \frac{\left(2 \times 10^{2}\right)^{3}\left(6 \times 10^{-2}\right)^{2}}{\left(2 \times 10^{-2}\right)\left(3 \times 10^{4}\right)^{2}}, \] seguimos los siguientes pasos: 1. **Calculamos el numerador:** - Primero, calculamos \((2 \times 10^{2})^{3}\): \[ (2^{3}) \times (10^{2 \times 3}) = 8 \times 10^{6}. \] - Luego, calculamos \((6 \times 10^{-2})^{2}\): \[ (6^{2}) \times (10^{-2 \times 2}) = 36 \times 10^{-4}. \] - Ahora multiplicamos ambos resultados: \[ (8 \times 10^{6}) \times (36 \times 10^{-4}) = 288 \times 10^{6 - 4} = 288 \times 10^{2}. \] 2. **Calculamos el denominador:** - Primero, calculamos \((3 \times 10^{4})^{2}\): \[ (3^{2}) \times (10^{4 \times 2}) = 9 \times 10^{8}. \] - Luego, multiplicamos esto por \((2 \times 10^{-2})\): \[ (2 \times 10^{-2}) \times (9 \times 10^{8}) = 18 \times 10^{-2 + 8} = 18 \times 10^{6}. \] 3. **Dividimos el numerador por el denominador:** \[ \frac{288 \times 10^{2}}{18 \times 10^{6}} = \frac{288}{18} \times 10^{2 - 6} = 16 \times 10^{-4}. \] 4. **Convertimos a forma decimal:** \[ 16 \times 10^{-4} = 0.0016. \] Por lo tanto, el resultado de la expresión es \[ 0.0016. \]