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

Answer 1

To simplify the expression \((4x^{-4}y^0)^{-2}\), we can follow these steps:

1. **Simplify inside the parentheses**:
The term \(y^0 = 1\) for any nonzero \(y\). Therefore, we can rewrite the expression as:
\[
(4x^{-4} \cdot 1)^{-2} = (4x^{-4})^{-2}
\]

2. **Apply the negative exponent rule**:
The rule states that \(a^{-n} = \frac{1}{a^n}\). Thus,
\[
(4x^{-4})^{-2} = \frac{1}{(4x^{-4})^2}
\]

3. **Apply the exponent to each factor**:
We raise each factor in the parentheses to the power of \(-2\):
\[
(4x^{-4})^2 = 4^2 \cdot (x^{-4})^2 = 16 \cdot x^{-8}
\]

4. **Combine the terms**:
Now substituting back, we have:
\[
\frac{1}{16x^{-8}} = \frac{1}{16} \cdot x^8
\]

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

In conclusion, the simplified expression is \(\frac{x^8}{16}\).