20) \( \left(\frac{x^{-7} y^{-2} z^{10}}{\left(x^{-1} y^{-2} z^{0} \cdot\left(x^{7} y^{8} z^{-3}\right)^{10}\right)^{9}}\right)^{10} \)

Simplify the expression by following steps: - step0: Solution: \(\left(\frac{x^{-7}y^{-2}z^{10}}{\left(\left(x^{-1}y^{-2}z^{0}\left(x^{7}y^{8}z^{-3}\right)^{10}\right)^{9}\right)}\right)^{10}\) - step1: Evaluate: \(\left(\frac{x^{-7}y^{-2}z^{10}}{\left(x^{-1}y^{-2}z^{0}\left(x^{7}y^{8}z^{-3}\right)^{10}\right)^{9}}\right)^{10}\) - step2: Evaluate the power: \(\left(\frac{x^{-7}y^{-2}z^{10}}{\left(x^{-1}y^{-2}\times 1\times \left(x^{7}y^{8}z^{-3}\right)^{10}\right)^{9}}\right)^{10}\) - step3: Multiply the terms: \(\left(\frac{x^{-7}y^{-2}z^{10}}{\left(\frac{x^{69}y^{78}}{z^{30}}\right)^{9}}\right)^{10}\) - step4: Divide the terms: \(\left(\frac{z^{280}}{x^{628}y^{704}}\right)^{10}\) - step5: Rewrite the expression: \(\frac{\left(z^{280}\right)^{10}}{\left(x^{628}y^{704}\right)^{10}}\) - step6: Evaluate the power: \(\frac{z^{2800}}{x^{6280}y^{7040}}\) To simplify the expression \[ \left(\frac{x^{-7} y^{-2} z^{10}}{\left(x^{-1} y^{-2} z^{0} \cdot\left(x^{7} y^{8} z^{-3}\right)^{10}\right)^{9}}\right)^{10}, \] we can break it down step by step. 1. **Simplify the denominator:** - The denominator is \(\left(x^{-1} y^{-2} z^{0} \cdot (x^{7} y^{8} z^{-3})^{10}\right)^{9}\). - First, simplify \((x^{7} y^{8} z^{-3})^{10}\): \[ (x^{7})^{10} = x^{70}, \quad (y^{8})^{10} = y^{80}, \quad (z^{-3})^{10} = z^{-30}. \] Thus, \[ (x^{7} y^{8} z^{-3})^{10} = x^{70} y^{80} z^{-30}. \] - Now, combine this with \(x^{-1} y^{-2} z^{0}\): \[ x^{-1} y^{-2} z^{0} \cdot x^{70} y^{80} z^{-30} = x^{69} y^{78} z^{-30}. \] - Now raise this to the 9th power: \[ (x^{69} y^{78} z^{-30})^{9} = x^{621} y^{702} z^{-270}. \] 2. **Combine the numerator and denominator:** - The entire expression now looks like: \[ \frac{x^{-7} y^{-2} z^{10}}{x^{621} y^{702} z^{-270}}. \] - This can be simplified by subtracting the exponents: \[ x^{-7 - 621} y^{-2 - 702} z^{10 - (-270)} = x^{-628} y^{-704} z^{280}. \] 3. **Raise the entire fraction to the 10th power:** - Now we raise the simplified expression to the 10th power: \[ (x^{-628} y^{-704} z^{280})^{10} = x^{-6280} y^{-7040} z^{2800}. \] 4. **Final result:** - The final simplified expression is: \[ \frac{z^{2800}}{x^{6280} y^{7040}}. \] Thus, the simplified form of the original expression is \[ \frac{z^{2800}}{x^{6280} y^{7040}}. \]