Use the rules of exponents to simplify the following expression. Assume that all variables are nonzero.

Question

left parenthesis 4 x Superscript negative 4 Baseline y Superscript nothing Baseline right parenthesis Superscript negative 2
Question content area bottom
Part 1
left parenthesis 4 x Superscript negative 4 Baseline y Superscript nothing Baseline right parenthesis Superscript negative 2equals
  
enter your response here
​(Simplify your answer. Use integers or fractions for any numbers in the expression. Use positive exponents​ only.)

Answer 1

To simplify the expression \((4x^{-4}y^{0})^{-2}\), we can follow the rules of exponents step by step.

1. Recall that \(y^{0} = 1\): \[ 4x^{-4}y^{0} = 4x^{-4} \cdot 1 = 4x^{-4} \]

2. Apply the negative exponent rule: The rule states that \((a^m)^{-n} = a^{-mn} = \frac{1}{a^{mn}}\).

   Thus, we have: \[ (4x^{-4})^{-2} = 4^{-2}(x^{-4})^{-2} \]

3. Simplify each component:

   • For \(4^{-2}\): \[ 4^{-2} = \frac{1}{4^2} = \frac{1}{16} \]
   • For \((x^{-4})^{-2}\): \[ (x^{-4})^{-2} = x^{8} \text{ (since } -4 \cdot -2 = 8\text{)} \]

4. Combine the results: \[ (4x^{-4})^{-2} = \frac{1}{16} \cdot x^{8} = \frac{x^8}{16} \]

Therefore, the simplified expression is: \[ \frac{x^8}{16} \]