a. \( 23^{18} \cdot\left(23^{5}\right)^{4}:\left(23^{30} \cdot 23^{7}\right) \cdot 23= \) b. \( 6^{58}:\left(6^{13}\right)^{3}: 6^{19} \cdot\left(6^{2}\right)^{2}= \) c. \( \left(10^{15}\right)^{10} \cdot 10^{124}:\left(10^{10}\right)^{20}: 10^{70}= \) d. \( \sqrt[5]{5} \cdot \sqrt[5]{5^{3}} \cdot \sqrt[5]{5^{4}}: \sqrt[5]{5^{3}}= \) e. \( \sqrt[6]{16} \cdot \sqrt[6]{8}: \sqrt[6]{2}= \) f. \( \sqrt{81 \cdot 64: 144}= \)

Calculate the value by following steps: - step0: Calculate: \(\sqrt{\frac{81\times 64}{144}}\) - step1: Reduce the fraction: \(\sqrt{36}\) - step2: Write in exponential form: \(\sqrt{6^{2}}\) - step3: Simplify the root: \(6\) Calculate or simplify the expression \( (10^(15))^(10) * 10^(124) / (10^(10))^(20) / 10^(70) \). Calculate the value by following steps: - step0: Calculate: \(\frac{\frac{\left(10^{15}\right)^{10}\times 10^{124}}{\left(10^{10}\right)^{20}}}{10^{70}}\) - step1: Multiply the exponents: \(\frac{\frac{\left(10^{15}\right)^{10}\times 10^{124}}{10^{10\times 20}}}{10^{70}}\) - step2: Multiply by \(a^{-n}:\) \(\frac{\left(10^{15}\right)^{10}\times 10^{124}\times 10^{-10\times 20}}{10^{70}}\) - step3: Multiply by \(a^{-n}:\) \(\left(10^{15}\right)^{10}\times 10^{124}\times 10^{-10\times 20}\times 10^{-70}\) - step4: Multiply the exponents: \(10^{15\times 10}\times 10^{124}\times 10^{-10\times 20}\times 10^{-70}\) - step5: Multiply the numbers: \(10^{150}\times 10^{124}\times 10^{-10\times 20}\times 10^{-70}\) - step6: Multiply the numbers: \(10^{150}\times 10^{124}\times 10^{-200}\times 10^{-70}\) - step7: Multiply the terms: \(10^{150+124-200-70}\) - step8: Calculate: \(10^{4}\) - step9: Evaluate the power: \(10000\) Calculate or simplify the expression \( (5^(1/5)) * (5^(3/5)) * (5^(4/5)) / (5^(3/5)) \). Calculate the value by following steps: - step0: Calculate: \(\frac{5^{\frac{1}{5}}\times 5^{\frac{3}{5}}\times 5^{\frac{4}{5}}}{5^{\frac{3}{5}}}\) - step1: Multiply by \(a^{-n}:\) \(5^{\frac{1}{5}}\times 5^{\frac{3}{5}}\times 5^{\frac{4}{5}}\times 5^{-\frac{3}{5}}\) - step2: Multiply the terms: \(5^{\frac{1}{5}+\frac{3}{5}+\frac{4}{5}-\frac{3}{5}}\) - step3: Calculate: \(5\) Calculate or simplify the expression \( (16^(1/6)) * (8^(1/6)) / (2^(1/6)) \). Calculate the value by following steps: - step0: Calculate: \(\frac{16^{\frac{1}{6}}\times 8^{\frac{1}{6}}}{2^{\frac{1}{6}}}\) - step1: Multiply: \(\frac{2^{\frac{7}{6}}}{2^{\frac{1}{6}}}\) - step2: Divide the numbers: \(2^{\frac{7}{6}-\frac{1}{6}}\) - step3: Subtract the terms: \(2^{1}\) - step4: Simplify: \(2\) Calculate or simplify the expression \( 23^(18) * (23^(5))^(4) / (23^(30) * 23^(7)) * 23 \). Calculate the value by following steps: - step0: Calculate: \(\frac{23^{18}\left(23^{5}\right)^{4}}{\left(23^{30}\times 23^{7}\right)}\times 23\) - step1: Remove the parentheses: \(\frac{23^{18}\left(23^{5}\right)^{4}}{23^{30}\times 23^{7}}\times 23\) - step2: Multiply by \(a^{-n}:\) \(23^{18}\left(23^{5}\right)^{4}\times 23^{-30}\times 23^{-7}\times 23\) - step3: Multiply the exponents: \(23^{18}\times 23^{5\times 4}\times 23^{-30}\times 23^{-7}\times 23\) - step4: Multiply the numbers: \(23^{18}\times 23^{20}\times 23^{-30}\times 23^{-7}\times 23\) - step5: Multiply the terms: \(23^{18+20-30-7+1}\) - step6: Calculate: \(23^{2}\) - step7: Evaluate the power: \(529\) Calculate or simplify the expression \( 6^(58) / (6^(13))^(3) / (6^(19) * (6^(2))^(2)) \). Calculate the value by following steps: - step0: Calculate: \(\frac{\frac{6^{58}}{\left(6^{13}\right)^{3}}}{\left(6^{19}\left(6^{2}\right)^{2}\right)}\) - step1: Remove the parentheses: \(\frac{\frac{6^{58}}{\left(6^{13}\right)^{3}}}{6^{19}\left(6^{2}\right)^{2}}\) - step2: Multiply the exponents: \(\frac{\frac{6^{58}}{\left(6^{13}\right)^{3}}}{6^{19}\times 6^{2\times 2}}\) - step3: Multiply the exponents: \(\frac{\frac{6^{58}}{6^{13\times 3}}}{6^{19}\times 6^{2\times 2}}\) - step4: Multiply by \(a^{-n}:\) \(\frac{6^{58}\times 6^{-13\times 3}}{6^{19}\times 6^{2\times 2}}\) - step5: Multiply by \(a^{-n}:\) \(6^{58}\times 6^{-13\times 3}\times 6^{-19}\times 6^{-2\times 2}\) - step6: Multiply the numbers: \(6^{58}\times 6^{-39}\times 6^{-19}\times 6^{-2\times 2}\) - step7: Multiply the numbers: \(6^{58}\times 6^{-39}\times 6^{-19}\times 6^{-4}\) - step8: Multiply the terms: \(6^{58-39-19-4}\) - step9: Subtract the numbers: \(6^{-4}\) - step10: Express with a positive exponent: \(\frac{1}{6^{4}}\) - step11: Evaluate the power: \(\frac{1}{1296}\) Aquí están las soluciones a los problemas planteados: a. \( 23^{18} \cdot\left(23^{5}\right)^{4}:\left(23^{30} \cdot 23^{7}\right) \cdot 23 = 529 \) b. \( 6^{58}:\left(6^{13}\right)^{3}: 6^{19} \cdot\left(6^{2}\right)^{2} = \frac{1}{1296} \) c. \( \left(10^{15}\right)^{10} \cdot 10^{124}:\left(10^{10}\right)^{20}: 10^{70} = 10000 \) d. \( \sqrt[5]{5} \cdot \sqrt[5]{5^{3}} \cdot \sqrt[5]{5^{4}}: \sqrt[5]{5^{3}} = 5 \) e. \( \sqrt[6]{16} \cdot \sqrt[6]{8}: \sqrt[6]{2} = 2 \) f. \( \sqrt{81 \cdot 64: 144} = 6 \)