5. Simplifica las siguientes expresiones. \( \begin{array}{ll}\text { a. } \frac{25^{3} \cdot\left(10^{3}\right)^{4} \cdot 169}{13 \cdot 125 \cdot 10^{9}} & \text { d. } \frac{10^{25} \cdot 495}{\left(210^{4}\right)^{2} \cdot 100^{5} \%} \\ \text { b. } \frac{\left[(15)^{2}\right]^{2} \cdot 81 \cdot 8}{108 \cdot 9^{-4} \cdot 225} & \text { e. }\left(-7 x^{4} y^{3}\right)^{2} \cdot\left(5 x^{2} 4\right) \\ \text { c. } \frac{30^{20} \cdot 15^{2}}{6^{14}} & \text { f. }\left(\frac{6 x^{4} y^{-2}}{12 x y^{3} z^{-1}}\right)^{-3}\end{array} \)

Calculate the value by following steps: - step0: Calculate: \(\frac{\left(30^{20}\times 15^{2}\right)}{6^{14}}\) - step1: Remove the parentheses: \(\frac{30^{20}\times 15^{2}}{6^{14}}\) - step2: Factor the expression: \(\frac{6^{20}\times 5^{20}\times 15^{2}}{6^{14}}\) - step3: Reduce the fraction: \(6^{6}\times 5^{20}\times 15^{2}\) - step4: Expand the expression: \(46656\times 5^{20}\times 15^{2}\) - step5: Multiply the terms: \(10497600\times 5^{20}\) Calculate or simplify the expression \( (10^25 * 495) / ((210^4)^2 * 100^5) \). Calculate the value by following steps: - step0: Calculate: \(\frac{\left(10^{25}\times 495\right)}{\left(\left(210^{4}\right)^{2}\times 100^{5}\right)}\) - step1: Remove the parentheses: \(\frac{10^{25}\times 495}{\left(210^{4}\right)^{2}\times 100^{5}}\) - step2: Multiply the exponents: \(\frac{10^{25}\times 495}{210^{4\times 2}\times 100^{5}}\) - step3: Multiply the numbers: \(\frac{10^{25}\times 495}{210^{8}\times 100^{5}}\) - step4: Factor the expression: \(\frac{10^{25}\times 495}{10^{8}\times 21^{8}\times 100^{5}}\) - step5: Reduce the fraction: \(\frac{10^{17}\times 495}{21^{8}\times 100^{5}}\) - step6: Factor the expression: \(\frac{10^{17}\times 495}{21^{8}\times 10^{10}}\) - step7: Reduce the fraction: \(\frac{10^{7}\times 495}{21^{8}}\) - step8: Factor the expression: \(\frac{10^{7}\times 9\times 55}{21^{8}}\) - step9: Factor the expression: \(\frac{10^{7}\times 9\times 55}{3^{8}\times 7^{8}}\) - step10: Reduce the fraction: \(\frac{10^{7}\times 55}{729\times 7^{8}}\) - step11: Calculate: \(\frac{55\times 10^{7}}{729\times 7^{8}}\) Calculate or simplify the expression \( ((6 * x^4 * y^(-2)) / (12 * x * y^3 * z^(-1)))^(-3) \). Simplify the expression by following steps: - step0: Solution: \(\left(\frac{\left(6x^{4}y^{-2}\right)}{\left(12xy^{3}z^{-1}\right)}\right)^{-3}\) - step1: Remove the parentheses: \(\left(\frac{6x^{4}y^{-2}}{12xy^{3}z^{-1}}\right)^{-3}\) - step2: Multiply by \(a^{-n}:\) \(\left(\frac{6x^{4}y^{-2}x^{-1}y^{-3}}{12z^{-1}}\right)^{-3}\) - step3: Reduce the fraction: \(\left(\frac{x^{3}y^{-5}}{2z^{-1}}\right)^{-3}\) - step4: Rewrite the expression: \(\left(\frac{x^{3}z}{2y^{5}}\right)^{-3}\) - step5: Transform the expression: \(\left(\frac{2y^{5}}{x^{3}z}\right)^{3}\) - step6: Rewrite the expression: \(\frac{\left(2y^{5}\right)^{3}}{\left(x^{3}z\right)^{3}}\) - step7: Evaluate the power: \(\frac{8y^{15}}{x^{9}z^{3}}\) Calculate or simplify the expression \( ((-7 * x^4 * y^3)^2 * (5 * x^2 * 4)) \). Simplify the expression by following steps: - step0: Solution: \(\left(-7x^{4}y^{3}\right)^{2}\left(5x^{2}\times 4\right)\) - step1: Remove the parentheses: \(\left(-7x^{4}y^{3}\right)^{2}\times 5x^{2}\times 4\) - step2: Multiply the terms: \(\left(-7x^{4}y^{3}\right)^{2}\times 20x^{2}\) - step3: Multiply the terms: \(980x^{8}y^{6}x^{2}\) - step4: Multiply the terms: \(980x^{10}y^{6}\) Calculate or simplify the expression \( (25^3 * (10^3)^4 * 169) / (13 * 125 * 10^9) \). Calculate the value by following steps: - step0: Calculate: \(\frac{\left(25^{3}\left(10^{3}\right)^{4}\times 169\right)}{\left(13\times 125\times 10^{9}\right)}\) - step1: Remove the parentheses: \(\frac{25^{3}\left(10^{3}\right)^{4}\times 169}{13\times 125\times 10^{9}}\) - step2: Multiply the exponents: \(\frac{25^{3}\times 10^{3\times 4}\times 169}{13\times 125\times 10^{9}}\) - step3: Multiply by \(a^{-n}:\) \(\frac{25^{3}\times 10^{3\times 4}\times 169\times 10^{-9}}{13\times 125}\) - step4: Multiply the numbers: \(\frac{25^{3}\times 10^{12}\times 169\times 10^{-9}}{13\times 125}\) - step5: Multiply: \(\frac{25^{3}\times 10^{3}\times 169}{13\times 125}\) - step6: Multiply the numbers: \(\frac{25^{3}\times 10^{3}\times 169}{1625}\) - step7: Reduce the fraction: \(\frac{25^{3}\times 10^{3}\times 13}{125}\) - step8: Factor the expression: \(\frac{5^{6}\times 10^{3}\times 13}{5^{3}}\) - step9: Reduce the fraction: \(5^{3}\times 10^{3}\times 13\) - step10: Multiply the numbers: \(50^{3}\times 13\) - step11: Multiply the terms: \(13\times 50^{3}\) Calculate or simplify the expression \( ([(15)^2]^2 * 81 * 8) / (108 * 9^(-4) * 225) \). Calculate the value by following steps: - step0: Calculate: \(\frac{\left(\left(15^{2}\right)^{2}\times 81\times 8\right)}{\left(108\times 9^{-4}\times 225\right)}\) - step1: Remove the parentheses: \(\frac{\left(15^{2}\right)^{2}\times 81\times 8}{108\times 9^{-4}\times 225}\) - step2: Multiply the exponents: \(\frac{15^{2\times 2}\times 81\times 8}{108\times 9^{-4}\times 225}\) - step3: Multiply the numbers: \(\frac{15^{4}\times 81\times 8}{108\times 9^{-4}\times 225}\) - step4: Multiply the terms: \(\frac{15^{4}\times 648}{108\times 9^{-4}\times 225}\) - step5: Multiply the terms: \(\frac{15^{4}\times 648}{24300\times 9^{-4}}\) - step6: Reduce the fraction: \(\frac{15^{4}\times 2}{75\times 9^{-4}}\) - step7: Factor the expression: \(\frac{15^{4}\times 2}{15\times 5\times 9^{-4}}\) - step8: Reduce the fraction: \(\frac{15^{3}\times 2}{5\times 9^{-4}}\) - step9: Factor the expression: \(\frac{5^{3}\times 3^{3}\times 2}{5\times 9^{-4}}\) - step10: Reduce the fraction: \(\frac{5^{2}\times 3^{3}\times 2}{9^{-4}}\) - step11: Factor the expression: \(\frac{5^{2}\times 3^{3}\times 2}{3^{-8}}\) - step12: Reduce the fraction: \(5^{2}\times 3^{11}\times 2\) - step13: Expand the expression: \(25\times 3^{11}\times 2\) - step14: Multiply the terms: \(50\times 3^{11}\) Aquí están las simplificaciones de las expresiones solicitadas: ### a. \[ \frac{25^{3} \cdot (10^{3})^{4} \cdot 169}{13 \cdot 125 \cdot 10^{9}} = 13 \times 50^{3} = 1625000 \] ### b. \[ \frac{[(15)^{2}]^{2} \cdot 81 \cdot 8}{108 \cdot 9^{-4} \cdot 225} = 50 \times 3^{11} = 8857350 \] ### c. \[ \frac{30^{20} \cdot 15^{2}}{6^{14}} = 10497600 \times 5^{20} \approx 1.001129 \times 10^{21} \] ### d. \[ \frac{10^{25} \cdot 495}{(210^{4})^{2} \cdot 100^{5}} = \frac{55 \times 10^{7}}{729 \times 7^{8}} \] ### e. \[ (-7 x^{4} y^{3})^{2} \cdot (5 x^{2} \cdot 4) = 980 x^{10} y^{6} \] ### f. \[ \left(\frac{6 x^{4} y^{-2}}{12 x y^{3} z^{-1}}\right)^{-3} = \frac{8 y^{15}}{x^{9} z^{3}} \] Si necesitas más detalles sobre algún paso específico, no dudes en preguntar.