Apply Laws of Exponents to write an equivalent expression for the given expression.
( – \frac{1}{3} w^{-4} z^{-2} )^{-3} = ( – \frac{1}{3} )^{-3} z^{12} (Simplify your answers.)
Apply Laws of Exponents to write an equivalent expression for the given expression. ( – \frac{1}{3} w^{-4} z^{-2} )^{-3} = ( – \frac{1}{3} )^{-3} z^{12} Simplify your answers
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To simplify the expression ((- frac{1}{3} w^{-4} z^{-2})^{-3}), we apply the laws of exponents.
1. Reciprocal and Power of a Power: When we raise a product to a negative exponent, we can take the reciprocal of the base and change the sign of the exponent:
[
(a cdot b)^{-n} = frac{1}{(a cdot b)^{n}}
]
Therefore:
[
( – frac{1}{3} w^{-4} z^{-2} )^{-3} = frac{1}{(- frac{1}{3} w^{-4} z^{-2})^{3}}
]
2. Distributing the Exponent: Now apply the exponent to each factor inside the parentheses:
[
= frac{1}{(-frac{1}{3})^{3} (w^{-4})^{3} (z^{-2})^{3}}
]
3. Calculating Each Factor:
– ((- frac{1}{3})^{3} = -frac{1}{27})
– ((w^{-4})^{3} = w^{-12}) (which can be written as frac{1}{w^{12}}))
– ((z^{-2})^{3} =