(1.) \( \frac{\left(x^{2} y^{3} z^{-6}\right)\left(a^{5} x^{4} z^{3}\right)}{\left(a^{2} x^{3} y^{2} z^{6}\right)} \) (2) \( \frac{\left(a^{-2} b^{5} c^{6} y^{9}\right)\left(a^{-4} b^{3} x^{6} z^{8}\right)}{\left(a^{6} c^{-3} x^{-4} z^{3}\right)} \) (3) \( \frac{\left(x^{6} y^{7} z^{8}\right)\left(x^{3} y^{2} z^{1}\right)\left(a^{6} b^{4}\right.}{\left(a^{2} b^{4} c^{6}\right)\left(x^{1} y^{2} z^{3}\right)\left(x^{8} y^{7}\right.} \)

To simplify the first expression \( \frac{\left(x^{2} y^{3} z^{-6}\right)\left(a^{5} x^{4} z^{3}\right)}{\left(a^{2} x^{3} y^{2} z^{6}\right)} \), you'll want to combine the terms in the numerator and then subtract the exponents of the same base in the denominator. Multiply the \( x \), \( y \), and \( z \) terms together and the \( a \) terms too. This will lead to \( a^{3} x^{3} y^{1} z^{-9} \) after combining and simplifying. Now for the second expression \( \frac{\left(a^{-2} b^{5} c^{6} y^{9}\right)\left(a^{-4} b^{3} x^{6} z^{8}\right)}{\left(a^{6} c^{-3} x^{-4} z^{3}\right)} \), again combine the numerators to get \( a^{-6} b^{8} c^{6} x^{6} y^{9} z^{8} \) and then perform exponent subtraction for each base on the denominator. You’ll find that the result simplifies to \( a^{-12} b^{8} c^{9} x^{10} y^{9} z^{5} \), which is quite elegant! Let the simplification phase inspire you—after all, simplifying expressions can truly make the math world sparkle, just like organizing a junk drawer can lead to exciting discoveries! And if you're eager to deepen your understanding, try experimenting with polynomial expressions or exponent rules through fun practical exercises like creating equations from everyday scenarios. Playing with variables could make you the next algebra magician!