1) \( \frac{\left(a^{2} b^{7}\right)^{6}}{\left(a^{2} b^{3}\right)^{4}}= \) 2) \( \frac{x^{2} x^{9} y^{3} y^{9} z^{5} z^{4}}{z^{3} y^{12} x^{30}} \) 3) \( \left(\frac{\left(a^{3} b^{4}\right)^{3}}{a^{4} a^{3} b b^{5}}\right)\left(b^{2} a^{9}\right) \)

Let's simplify each expression step by step. 1) For the expression \( \frac{\left(a^{2} b^{7}\right)^{6}}{\left(a^{2} b^{3}\right)^{4}} \): First, we simplify the numerator and the denominator separately. - **Numerator**: \[ \left(a^{2} b^{7}\right)^{6} = a^{2 \cdot 6} b^{7 \cdot 6} = a^{12} b^{42} \] - **Denominator**: \[ \left(a^{2} b^{3}\right)^{4} = a^{2 \cdot 4} b^{3 \cdot 4} = a^{8} b^{12} \] Now, we can write the expression as: \[ \frac{a^{12} b^{42}}{a^{8} b^{12}} \] Using the properties of exponents, we can simplify: \[ = a^{12 - 8} b^{42 - 12} = a^{4} b^{30} \] So, the final result for the first expression is: \[ \boxed{a^{4} b^{30}} \] --- 2) For the expression \( \frac{x^{2} x^{9} y^{3} y^{9} z^{5} z^{4}}{z^{3} y^{12} x^{30}} \): First, we simplify the numerator: \[ x^{2} x^{9} = x^{2 + 9} = x^{11} \] \[ y^{3} y^{9} = y^{3 + 9} = y^{12} \] \[ z^{5} z^{4} = z^{5 + 4} = z^{9} \] So, the numerator becomes: \[ x^{11} y^{12} z^{9} \] Now, we can write the expression as: \[ \frac{x^{11} y^{12} z^{9}}{z^{3} y^{12} x^{30}} \] Now, simplify the denominator: \[ = z^{3} y^{12} x^{30} \] Now we can simplify: \[ = \frac{x^{11}}{x^{30}} \cdot \frac{y^{12}}{y^{12}} \cdot \frac{z^{9}}{z^{3}} = x^{11 - 30} y^{12 - 12} z^{9 - 3} = x^{-19} y^{0} z^{6} \] Since \( y^{0} = 1 \), we can write: \[ = \frac{z^{6}}{x^{19}} \] So, the final result for the second expression is: \[ \boxed{\frac{z^{6}}{x^{19}}} \] --- 3) For the expression \( \left(\frac{\left(a^{3} b^{4}\right)^{3}}{a^{4} a^{3} b b^{5}}\right)\left(b^{2} a^{9}\right) \): First, simplify the fraction: - **Numerator**: \[ \left(a^{3} b^{4}\right)^{3} = a^{3 \cdot 3} b^{4 \cdot 3} = a^{9} b^{12} \] - **Denominator**: \[ a^{4} a^{3} b b^{5} = a^{4 + 3} b^{1 + 5} = a^{7} b^{6} \] Now we can write the expression as: \[ \frac{a^{9} b^{12}}{a^{7} b^{6}} = a^{9 - 7} b^{12 - 6} = a^{2} b^{6} \] Now, we multiply by \( \left(b^{2} a^{9}\right) \): \[ (a^{2} b^{6})(b^{2} a^{9}) = a^{2 + 9} b^{6 + 2} = a^{11} b^{8} \] So, the final result for the third expression is: \[ \boxed{a^{11} b^{8}} \]