( – frac{1}{3} w^{-3} z^{-1} )^{-2} = ( – frac{1}{3} )^{-2} square z^{6} square

To solve the equation ((- frac{1}{3} w^{-3} z^{-1})^{-2} = (- frac{1}{3})^{-2} z^{6}), we can start by applying the exponent to each term inside the parentheses.

1. Apply the exponent to each part of the left side:

[

(- frac{1}{3})^{-2} cdot (w^{-3})^{-2} cdot (z^{-1})^{-2}

]

2. Calculate each part:

– ((- frac{1}{3})^{-2} = frac{1}{(-frac{1}{3})^2} = frac{1}{frac{1}{9}} = 9)

– ((w^{-3})^{-2} = w^{6})

– ((z^{-1})^{-2} = z^{2})

Combining these results gives us:

[

9 w^{6} z^{2}

]

3. Set the left side equal to the right side:

[

9 w^{6} z^{2} = frac{1}{9} z^{6}

]

4. Rearranging gives:

– On the right side, (frac{1}{9} z^{6} = 9 w^{6} z