- Simplify the expression inside the radical: \(\frac{10x}{3x^3} = \frac{10}{3x^2}\).
- Rationalize the denominator by multiplying the numerator and denominator by \(\sqrt[5]{3^4x^3}\).
- The resulting expression is \(\frac{\sqrt[5]{10 \cdot 81 \cdot x^3}}{3x} = \frac{\sqrt[5]{810x^3}}{3x}\).
- The final simplified form is \(\boxed{\frac{\sqrt[5]{810x^3}}{3x}}\).
Explanation
- Simplifying the inner fraction
The expression we need to simplify is \(\sqrt[5]{\frac{10x}{3x^3}}\) for \(x
eq 0\). To begin, we simplify the fraction inside the radical by canceling out a common factor of \(x\) from the numerator and the denominator:
\[\frac{10x}{3x^3} = \frac{10}{3x^2}\]
Thus, the expression becomes \(\sqrt[5]{\frac{10}{3x^2}}\).
- Rationalizing the denominator
To simplify the radical further, we aim to rationalize the denominator. The current denominator is \(\sqrt[5]{3x^2} = \sqrt[5]{3^1 x^2}\). For the root of a perfect fifth power to be easily extracted, the powers of the factors inside the root should be multiples of 5. To achieve this, we multiply the numerator and the denominator by \(\sqrt[5]{3^4 x^3}\) because:
\[\sqrt[5]{3^1 x^2} \times \sqrt[5]{3^4 x^3} = \sqrt[5]{3^{1+4} x^{2+3}} = \sqrt[5]{3^5 x^5} = 3x\]
This rationalization removes the radical from the denominator.
- Finalizing the simplification
Next, we multiply the numerator by the same expression \(\sqrt[5]{3^4 x^3}\):
\[\sqrt[5]{10} \times \sqrt[5]{3^4 x^3} = \sqrt[5]{10 \times 3^4 \times x^3}\]
Since \(3^4 = 81\), the numerator becomes:
\[\sqrt[5]{10 \times 81 \times x^3} = \sqrt[5]{810 x^3}\]
Combining this with the rationalized denominator obtained in the previous step, the simplified expression is:
\[\frac{\sqrt[5]{810 x^3}}{3x}\]
Comparing this to the given options, it matches the fourth option.
Examples
In engineering, when dealing with complex dimensions or unit conversions involving fifth roots (often appearing in certain fluid dynamic formulas or growth models), rationalizing denominators is a crucial step. It allows for easier calculations, numerical approximation, and comparison between different measurement values. Being able to manipulate these radicals helps maintain precision in scientific calculations and ensures that expressions are in their simplest, most standard form for further analysis.