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

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 \(\frac{x^8}{16}\).