lify: \( \left(\frac{1447^{\frac{-3}{4}} 8^{\frac{2}{7}} \sqrt[6]{x^{5}} \sqrt[5]{y^{3}} \sqrt[7]{z}}{12^{12} 8^{12} 7^{2} x^{\frac{4}{5}} \sqrt[3]{y^{5}} z^{\frac{2}{6}}} * \frac{x^{\frac{-3}{4}} y^{\frac{2}{7}} z^{\frac{-8}{5}} s^{\frac{3}{2}}}{x^{2} y^{5} z^{\frac{2}{6}} s^{-9}}\right)^{\frac{7}{4}} \)

Simplify the expression by following steps: - step0: Solution: \(\left(\frac{1447^{\frac{-3}{4}}\times 8^{\frac{2}{7}}\sqrt[6]{x^{5}}\times \sqrt[5]{y^{3}}\times \sqrt[7]{z}}{12^{12}\times 8^{12}\times 7^{2}x^{\frac{4}{5}}\sqrt[3]{y^{5}}\times z^{\frac{2}{6}}}\times \frac{x^{\frac{-3}{4}}y^{\frac{2}{7}}z^{\frac{-8}{5}}s^{\frac{3}{2}}}{x^{2}y^{5}z^{\frac{2}{6}}s^{-9}}\right)^{\frac{7}{4}}\) - step1: Multiply by \(a^{-n}:\) \(\left(\frac{1447^{\frac{-3}{4}}\times 8^{\frac{2}{7}}\sqrt[6]{x^{5}}\times \sqrt[5]{y^{3}}\times \sqrt[7]{z}}{12^{12}\times 8^{12}\times 7^{2}x^{\frac{4}{5}}\sqrt[3]{y^{5}}\times z^{\frac{2}{6}}}\times x^{\frac{-3}{4}}y^{\frac{2}{7}}z^{\frac{-8}{5}}s^{\frac{3}{2}}x^{-2}y^{-5}z^{-\frac{2}{6}}s^{9}\right)^{\frac{7}{4}}\) - step2: Multiply by \(a^{-n}:\) \(\left(\frac{1447^{\frac{-3}{4}}\times 8^{\frac{2}{7}}\sqrt[6]{x^{5}}\times \sqrt[5]{y^{3}}\times \sqrt[7]{z}\times 8^{-12}}{12^{12}\times 7^{2}x^{\frac{4}{5}}\sqrt[3]{y^{5}}\times z^{\frac{2}{6}}}\times x^{\frac{-3}{4}}y^{\frac{2}{7}}z^{\frac{-8}{5}}s^{\frac{3}{2}}x^{-2}y^{-5}z^{-\frac{2}{6}}s^{9}\right)^{\frac{7}{4}}\) - step3: Rewrite the fraction: \(\left(\frac{1447^{-\frac{3}{4}}\times 8^{\frac{2}{7}}\sqrt[6]{x^{5}}\times \sqrt[5]{y^{3}}\times \sqrt[7]{z}\times 8^{-12}}{12^{12}\times 7^{2}x^{\frac{4}{5}}\sqrt[3]{y^{5}}\times z^{\frac{2}{6}}}\times x^{\frac{-3}{4}}y^{\frac{2}{7}}z^{\frac{-8}{5}}s^{\frac{3}{2}}x^{-2}y^{-5}z^{-\frac{2}{6}}s^{9}\right)^{\frac{7}{4}}\) - step4: Reduce the fraction: \(\left(\frac{1447^{-\frac{3}{4}}\times 8^{\frac{2}{7}}\sqrt[6]{x^{5}}\times \sqrt[5]{y^{3}}\times \sqrt[7]{z}\times 8^{-12}}{12^{12}\times 7^{2}x^{\frac{4}{5}}\sqrt[3]{y^{5}}\times z^{\frac{1}{3}}}\times x^{\frac{-3}{4}}y^{\frac{2}{7}}z^{\frac{-8}{5}}s^{\frac{3}{2}}x^{-2}y^{-5}z^{-\frac{2}{6}}s^{9}\right)^{\frac{7}{4}}\) - step5: Simplify the root: \(\left(\frac{1447^{-\frac{3}{4}}\times 8^{\frac{2}{7}}\sqrt[6]{x^{5}}\times \sqrt[5]{y^{3}}\times \sqrt[7]{z}\times 8^{-12}}{12^{12}\times 7^{2}x^{\frac{4}{5}}y\sqrt[3]{y^{2}}\times z^{\frac{1}{3}}}\times x^{\frac{-3}{4}}y^{\frac{2}{7}}z^{\frac{-8}{5}}s^{\frac{3}{2}}x^{-2}y^{-5}z^{-\frac{2}{6}}s^{9}\right)^{\frac{7}{4}}\) - step6: Multiply: \(\left(\frac{\frac{\sqrt[420]{1447^{105}\times 2^{360}x^{350}y^{252}z^{60}}}{2894\times 2^{35}}}{12^{12}\times 7^{2}x^{\frac{4}{5}}y\sqrt[3]{y^{2}}\times z^{\frac{1}{3}}}\times x^{\frac{-3}{4}}y^{\frac{2}{7}}z^{\frac{-8}{5}}s^{\frac{3}{2}}x^{-2}y^{-5}z^{-\frac{2}{6}}s^{9}\right)^{\frac{7}{4}}\) - step7: Multiply the terms: \(\left(\frac{\frac{\sqrt[420]{1447^{105}\times 2^{360}x^{350}y^{252}z^{60}}}{2894\times 2^{35}}}{49\times 12^{12}y\sqrt[15]{y^{10}x^{12}z^{5}}}\times x^{\frac{-3}{4}}y^{\frac{2}{7}}z^{\frac{-8}{5}}s^{\frac{3}{2}}x^{-2}y^{-5}z^{-\frac{2}{6}}s^{9}\right)^{\frac{7}{4}}\) - step8: Rewrite the fraction: \(\left(\frac{\frac{\sqrt[420]{1447^{105}\times 2^{360}x^{350}y^{252}z^{60}}}{2894\times 2^{35}}}{49\times 12^{12}y\sqrt[15]{y^{10}x^{12}z^{5}}}\times x^{-\frac{3}{4}}y^{\frac{2}{7}}z^{\frac{-8}{5}}s^{\frac{3}{2}}x^{-2}y^{-5}z^{-\frac{2}{6}}s^{9}\right)^{\frac{7}{4}}\) - step9: Rewrite the fraction: \(\left(\frac{\frac{\sqrt[420]{1447^{105}\times 2^{360}x^{350}y^{252}z^{60}}}{2894\times 2^{35}}}{49\times 12^{12}y\sqrt[15]{y^{10}x^{12}z^{5}}}\times x^{-\frac{3}{4}}y^{\frac{2}{7}}z^{-\frac{8}{5}}s^{\frac{3}{2}}x^{-2}y^{-5}z^{-\frac{2}{6}}s^{9}\right)^{\frac{7}{4}}\) - step10: Reduce the fraction: \(\left(\frac{\frac{\sqrt[420]{1447^{105}\times 2^{360}x^{350}y^{252}z^{60}}}{2894\times 2^{35}}}{49\times 12^{12}y\sqrt[15]{y^{10}x^{12}z^{5}}}\times x^{-\frac{3}{4}}y^{\frac{2}{7}}z^{-\frac{8}{5}}s^{\frac{3}{2}}x^{-2}y^{-5}z^{-\frac{1}{3}}s^{9}\right)^{\frac{7}{4}}\) - step11: Divide the terms: \(\left(\frac{\sqrt[420]{1447^{105}\times 2^{360}x^{350}y^{252}z^{60}}}{141806\times 2^{35}\times 12^{12}y\sqrt[15]{y^{10}x^{12}z^{5}}}\times x^{-\frac{3}{4}}y^{\frac{2}{7}}z^{-\frac{8}{5}}s^{\frac{3}{2}}x^{-2}y^{-5}z^{-\frac{1}{3}}s^{9}\right)^{\frac{7}{4}}\) - step12: Multiply: \(\left(\frac{s^{10}\sqrt[420]{1447^{105}\times 2^{360}x^{119}y^{92}s^{210}}}{141806\times 2^{35}\times 12^{12}y^{6}z^{2}\sqrt[105]{z^{13}}\times x^{3}}\right)^{\frac{7}{4}}\) - step13: Rewrite the expression: \(\frac{\left(s^{10}\sqrt[420]{1447^{105}\times 2^{360}x^{119}y^{92}s^{210}}\right)^{\frac{7}{4}}}{\left(141806\times 2^{35}\times 12^{12}y^{6}z^{2}\sqrt[105]{z^{13}}\times x^{3}\right)^{\frac{7}{4}}}\) - step14: Evaluate the power: \(\frac{s^{18}\sqrt[240]{1447^{105}\times 2^{360}s^{90}y^{92}x^{119}}}{y^{10}x^{5}z^{3}\sqrt[60]{141806^{105}\times 2^{3675}\times 12^{1260}y^{30}x^{15}z^{43}}}\) - step15: Simplify: \(\frac{\frac{s^{18}x\sqrt[240]{1447^{105}\times 2^{360}z^{68}x^{59}y^{212}s^{90}}}{\sqrt[4]{\left(141806\times 2^{35}\times 12^{12}\right)^{3}}}}{70903\times 2^{60}\times 3^{12}y^{11}x^{6}z^{4}}\) - step16: Calculate: \(\frac{\frac{s^{18}x\sqrt[240]{1447^{105}\times 2^{360}z^{68}x^{59}y^{212}s^{90}}}{\sqrt[4]{141806^{3}\times 2^{105}\times 12^{36}}}}{70903\times 2^{60}\times 3^{12}y^{11}x^{6}z^{4}}\) - step17: Multiply by the reciprocal: \(\frac{s^{18}x\sqrt[240]{1447^{105}\times 2^{360}z^{68}x^{59}y^{212}s^{90}}}{\sqrt[4]{141806^{3}\times 2^{105}\times 12^{36}}}\times \frac{1}{70903\times 2^{60}\times 3^{12}y^{11}x^{6}z^{4}}\) - step18: Reduce the fraction: \(\frac{s^{18}\sqrt[240]{1447^{105}\times 2^{360}z^{68}x^{59}y^{212}s^{90}}}{\sqrt[4]{141806^{3}\times 2^{105}\times 12^{36}}}\times \frac{1}{70903\times 2^{60}\times 3^{12}y^{11}x^{5}z^{4}}\) - step19: Multiply the terms: \(\frac{s^{18}\sqrt[240]{1447^{105}\times 2^{360}z^{68}x^{59}y^{212}s^{90}}}{\sqrt[4]{141806^{3}\times 2^{105}\times 12^{36}}\times 70903\times 2^{60}\times 3^{12}y^{11}x^{5}z^{4}}\) - step20: Multiply the terms: \(\frac{s^{18}\sqrt[240]{1447^{105}\times 2^{360}z^{68}x^{59}y^{212}s^{90}}}{70903\times 2^{60}\times 3^{12}\sqrt[4]{141806^{3}\times 2^{105}\times 12^{36}}\times y^{11}x^{5}z^{4}}\) To simplify the expression \[ \left(\frac{1447^{\frac{-3}{4}} 8^{\frac{2}{7}} \sqrt[6]{x^{5}} \sqrt[5]{y^{3}} \sqrt[7]{z}}{12^{12} 8^{12} 7^{2} x^{\frac{4}{5}} \sqrt[3]{y^{5}} z^{\frac{2}{6}}} * \frac{x^{\frac{-3}{4}} y^{\frac{2}{7}} z^{\frac{-8}{5}} s^{\frac{3}{2}}}{x^{2} y^{5} z^{\frac{2}{6}} s^{-9}}\right)^{\frac{7}{4}}, \] we follow these steps: 1. **Combine the two fractions** in the expression. 2. **Simplify the powers** of each variable and constant. 3. **Apply the exponent** of \(\frac{7}{4}\) to the entire expression. After performing the calculations, the simplified result is: \[ \frac{s^{18}\sqrt[240]{1447^{105}\times 2^{360}z^{68}x^{59}y^{212}s^{90}}}{70903\times 2^{60}\times 3^{12}\sqrt[4]{141806^{3}\times 2^{105}\times 12^{36}}\times y^{11}x^{5}z^{4}}. \] This expression represents the fully simplified form of the original problem.