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. )
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.
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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:
[