- Simplify the fraction \(\frac{24}{128}\) to \(\frac{3}{16}\).
- Use exponent rules to simplify the variable part \(\frac{x^6 y}{x^4 y^5}\) to \(\frac{x^2}{y^4}\).
- Take the fourth root of \(\frac{3 x^2}{16 y^4}\) to obtain \(\frac{\sqrt[4]{3 x^2}}{2 y}\).
- The correct equivalent expression is \(\frac{\sqrt[4]{3 x^2}}{2 y}\).
- Final Answer: \(\boxed{\frac{\sqrt[4]{3 x^2}}{2 y}}\)
Explanation
- Simplifying the Fraction inside the Radical
We want to simplify the expression \(\sqrt[4]{\frac{24 x^6 y}{128 x^4 y^5}}\), where \(x
eq 0\) and \(y > 0\). We can begin by simplifying the fraction inside the radical.
First, divide the numerical coefficients:
\[\frac{24}{128} = \frac{3 \times 8}{16 \times 8} = \frac{3}{16}\]
Next, apply the rules of exponents for the variables \(x\) and \(y\):
\[\frac{x^6}{x^4} = x^{6-4} = x^2\]
\[\frac{y}{y^5} = y^{1-5} = y^{-4} = \frac{1}{y^4}\]
Combining these parts, the fraction inside the fourth root simplifies to:
\[\frac{24 x^6 y}{128 x^4 y^5} = \frac{3 x^2}{16 y^4}\]
- Evaluating the Fourth Root
Now we compute the fourth root of the simplified expression:
\[\sqrt[4]{\frac{3 x^2}{16 y^4}}\]
Using the property of radicals \(\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}\), we can split this into:
\[\frac{\sqrt[4]{3 x^2}}{\sqrt[4]{16 y^4}}\]
Next, simplify the denominator. Since \(\sqrt[4]{16} = 2\) and \(\sqrt[4]{y^4} = |y|\), and given that \(y > 0\), the denominator simplifies to \(2y\).
The expression becomes:
\[\frac{\sqrt[4]{3 x^2}}{2 y}\]
Comparing this result to the given options, it matches the fourth choice exactly.
Examples
This type of radical simplification is useful in fields like engineering and physics when dealing with formulas containing exponents and roots, such as calculating the radius of a sphere given its volume or in dimensional analysis. By simplifying expressions early, you reduce the complexity of the equations, making them much easier to work with when solving real-world problems involving physical dimensions or growth rates.