Question

Use the rules of exponents to simplify the following expression. Assume that all variables are nonzero.
left parenthesis 4 x Superscript negative 4 Baseline y Superscript nothing Baseline right parenthesis Superscript negative 2
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Part 1
left parenthesis 4 x Superscript negative 4 Baseline y Superscript nothing Baseline right parenthesis Superscript negative 2equals
  
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​(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:
\[
\frac{x^8}{16}
\]