Simplify ( (2xy^{-2}z^{4}) / (3xyz^{-1}) )^{-2} , where x, y, and z are not zero.

To simplify (left( frac{2xy^{-2}z^{4}}{3xyz^{-1}} right)^{-2}), we first simplify the fraction inside the parentheses.

1. Cancel out common terms:

– The (x) terms cancel: (x/x = 1).

– For (y), we have (y^{-2} / y = y^{-3}) (which is (1/y^{3})).

– For (z), (z^{4} / z^{-1} = z^{5}) (since (z^{4} cdot z^{1} = z^{5})).

So, we can rewrite the expression as:

[

frac{2}{3} cdot frac{1}{y^{3}} cdot z^{5} = frac{2z^{5}}{3y^{3}}

]

2. Apply the exponent -2:

Now, we raise the whole fraction to the power of -2:

[

left(frac{2z^{5}}{3y^{3}}right)^{-2} = frac{(3y^{3})^{2}}{(2z^{5})^{2}} = frac{9y^{6}}{4z^{10}}

]

Thus, the simplified expression is: