- Apply the negative exponent rule: \(\left(\frac{x}{y}\right)^{-n} = \left(\frac{y}{x}\right)^n\).
- Simplify the fraction: \(\left(\frac{1}{4ab}\right)^{-2} = (4ab)^2\).
- Expand the squared term using the power of a product rule: \((4ab)^2 = 16a^2b^2\).
- The final simplified expression is \(\boxed{16a^2b^2}\).
Explanation
- Understanding the Problem and Exponent Properties
We are asked to simplify the expression \(\left(\frac{1}{4ab}\right)^{-2}\) given that \(a
eq 0\) and \(b
eq 0\). The expression involves a power with a negative exponent, so we should recall the relevant properties of exponents. Specifically, the negative exponent property states that \(\left(\frac{x}{y}\right)^{-n} = \left(\frac{y}{x}\right)^n\) for any non-zero \(x\) and \(y\).
- Rewriting with a Positive Exponent
Using the property \(\left(\frac{x}{y}\right)^{-n} = \left(\frac{y}{x}\right)^n\), we can rewrite the original expression \(\left(\frac{1}{4ab}\right)^{-2}\) by taking the reciprocal of the base and changing the sign of the exponent to positive. This gives us:
\[\left(\frac{1}{4ab}\right)^{-2} = \left(\frac{4ab}{1}\right)^2 = (4ab)^2\]
- Expanding the Expression
Now we apply the power of a product rule, which states that \((xyz)^n = x^n y^n z^n\). Applying this to our expression \((4ab)^2\), we get:
\[(4ab)^2 = 4^2 \times a^2 \times b^2\]
Calculating the square of the constant 4 gives \(4^2 = 16\). Therefore:
\[(4ab)^2 = 16a^2b^2\]
- Comparison and Conclusion
Comparing our result \(16a^2b^2\) with the given options, we find it matches the fourth choice:
- \(-\frac{1}{16a^2b^2}\)
- \(\frac{a^2b^2}{4}\)
- \(-16a^2b^2\)
- \(16a^2b^2\)
The simplified form is \(16a^2b^2\), so the correct answer is \(\boxed{16a^2b^2}\).
Examples
This expression simplification is common in physics and engineering, such as calculating the area of a surface or the square of physical constants in formulas. For instance, if you have a physical scaling factor \(k = (4ab)^{-1}\) and need to find the inverse-square of that factor, you are essentially calculating \((k^{-1})^2 = (4ab)^2\), which is a common step in normalizing data or calculating energy flux where quantities depend on the square of inverse dimensions.