Apply Laws of Exponents to write an equivalent expression for the given expression: ( – 1/3 w ^ -4 z ^ -2 ) ^ -3. Simplify your answers. Simplify your answers.

To simplify the expression ((- frac{1}{3} w^{-4} z^{-2})^{-3}), we can apply the laws of exponents step by step.

1. Apply the Negative Exponent Rule: When you have a negative exponent, you can take the reciprocal of the base and change the sign of the exponent. Thus, we have:

[

(- frac{1}{3} w^{-4} z^{-2})^{-3} = frac{1}{(- frac{1}{3} w^{-4} z^{-2})^3}

]

2. Simplify the Base: Compute ((- frac{1}{3})^3), which equals (-frac{1}{27}), and for the variables, apply the power of a power rule ((a^m)^n = a^{m cdot n}):

[

w^{-4 cdot -3} = w^{12} quad text{and} quad z^{-2 cdot -3} = z^{6}

]

Thus:

[

(- frac{1}{3})^{-3} w^{12} z^{6} = -frac{1}{27} w^{12} z^{6}

]

3. Combine Everything Together:

Therefore, our final simplified expression is:

[