- Apply the power of a product rule to distribute the outer exponent \(-2\).
- Use the power of a power rule \((a^m)^n = a^{mn}\) to simplify each individual factor's exponent: \(x^{-1}\), \(y^{\frac{1}{2}}\), and \(z^{-2}\).
- Rewrite with positive exponents by applying \(a^{-n} = \frac{1}{a^n}\).
- Obtain the simplified expression \(\frac{y^{\frac{1}{2}}}{x z^2} = \boxed{\frac{\sqrt{y}}{x z^2}}\).
Explanation
- Analyzing the Expression and Identifying Relevant Laws of Exponents
We are given the expression \(\left(x^{\frac{1}{2}} y^{-\frac{1}{4}} z\right)^{-2}\) and tasked with simplifying it. The expression involves variables \(x, y,\) and \(z\) raised to fractional powers, and the entire product is raised to the power of \(-2\). To simplify this, we need to use the fundamental rules of exponents: the power of a product rule \((abc)^n = a^n b^n c^n\) and the power of a power rule \((a^m)^n = a^{mn}\).
- Applying Exponent Rules to Simplify Factors
First, apply the power of a product rule to distribute the exponent \(-2\) to each factor inside the parenthesis:
\[\left(x^{\frac{1}{2}} y^{-\frac{1}{4}} z\right)^{-2} = \left(x^{\frac{1}{2}}\right)^{-2} \cdot \left(y^{-\frac{1}{4}}\right)^{-2} \cdot (z)^{-2}.\]
Next, apply the power of a power rule \((a^m)^n = a^{mn}\) to each term:
- For \(x\): \(\left(x^{\frac{1}{2}}\right)^{-2} = x^{\frac{1}{2} \cdot (-2)} = x^{-1}\).
- For \(y\): \(\left(y^{-\frac{1}{4}}\right)^{-2} = y^{-\frac{1}{4} \cdot (-2)} = y^{\frac{2}{4}} = y^{\frac{1}{2}}\).
- For \(z\): \((z)^{-2} = z^{-2}\).
- Final Simplification and Conclusion
Now, combine the simplified factors:
\[x^{-1} \cdot y^{\frac{1}{2}} \cdot z^{-2}.\]
Using the definition of negative exponents \(a^{-n} = \frac{1}{a^n}\), we can rewrite this expression to have positive exponents:
\[x^{-1} = \frac{1}{x}, \quad z^{-2} = \frac{1}{z^2}.\]
Thus, the expression becomes:
\[\frac{y^{\frac{1}{2}}}{x \cdot z^2}.\]
Recalling that \(y^{\frac{1}{2}}\) is equivalent to the square root of \(y\) (\(\sqrt{y}\)), the final simplified form is \(\frac{\sqrt{y}}{x z^2}\). While none of the initially listed options exactly match this simplified form, the simplification process is mathematically rigorous based on the given expression.
Examples
In data analysis and scientific calculations, researchers often work with formulas involving dimensions or units raised to fractional powers (such as square roots for surface area or cube roots for volume). Simplifying expressions using exponent laws allows for easier computation and more efficient formula manipulation, helping scientists and engineers predict the behavior of complex systems like heat diffusion or fluid flow. Understanding how to handle these exponents ensures that complex physical models are correctly reduced to their most manageable forms.