To simplify the expression \(\left(4x^{3}y^{-4}\right)^{-2}\), we will apply the property of exponents that states \((a^m)^{-n} = a^{-mn}\) and the rule that states \((ab)^n = a^n b^n\).
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Distribute the exponent \(-2\) to each factor inside the parentheses: \[ \left(4x^{3}y^{-4}\right)^{-2} = 4^{-2} \cdot (x^{3})^{-2} \cdot (y^{-4})^{-2} \]
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Now simplify each part:
- \(4^{-2} = \frac{1}{4^2} = \frac{1}{16}\)
- \((x^{3})^{-2} = x^{-6}\)
- \((y^{-4})^{-2} = y^{8}\) (since \(-4 \cdot -2 = 8\))
Putting these together, we have: \[ \left(4x^{3}y^{-4}\right)^{-2} = \frac{1}{16} \cdot x^{-6} \cdot y^{8} \]
- Finally, rewrite the expression in a more standard form: \[ \frac{y^{8}}{16x^{6}} \]
Thus, the simplified expression is: \[ \frac{y^{8}}{16x^{6}} \]