- Simplify the fraction inside the root to \(\frac{x^5}{9}\).
- Use the quotient rule to separate the square root of the numerator and the denominator, resulting in \(\frac{\sqrt{x^5}}{3}\).
- Simplify the variable component in the numerator to \(x^2\sqrt{x}\).
- Combine the results to get \(\frac{x^2 \sqrt{x}}{3}\), which matches the first option.
- The final answer is \(\boxed{\frac{x^2 \sqrt{x}}{3}}\).
Explanation
- Simplify the fraction inside the square root
To simplify the expression \(\sqrt{\frac{2x^5}{18}}\) given that \(x \geq 0\), we start by analyzing the fraction inside the square root. We can reduce the fraction \(\frac{2}{18}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 2. This gives us \(\frac{2 \div 2}{18 \div 2} = \frac{1}{9}\). Therefore, the expression simplifies to \(\sqrt{\frac{x^5}{9}}\).
- Apply the square root quotient property
Next, we apply the property of square roots that states \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\) for \(a \geq 0\) and \(b > 0\). Using this property, our expression becomes:
\[\sqrt{\frac{x^5}{9}} = \frac{\sqrt{x^5}}{\sqrt{9}}\]
We know that \(\sqrt{9} = 3\), so the expression becomes \(\frac{\sqrt{x^5}}{3}\).
- Simplify the variable term in the numerator
Now, we simplify the term in the numerator, \(\sqrt{x^5}\). Since \(x^5 = x^4 \cdot x\), we can use the product property of square roots:
\[\sqrt{x^5} = \sqrt{x^4 \cdot x} = \sqrt{x^4} \cdot \sqrt{x}\]
Since \(x \geq 0\), \(\sqrt{x^4} = x^2\). Therefore, \(\sqrt{x^5} = x^2 \sqrt{x}\).
- Final calculation and comparison
Finally, we combine the simplified numerator and denominator:
\[\frac{\sqrt{x^5}}{3} = \frac{x^2 \sqrt{x}}{3}\]
Comparing this result with the given choices, we see that it matches the first option: \(\frac{x^2 \sqrt{x}}{3}\).
Examples
Simplifying radical expressions is fundamental in various fields, such as physics and engineering, where you might need to calculate magnitudes of forces or lengths in complex systems. For instance, if you are determining the distance of a particle from the origin in a coordinate system where its position is given as a power of time, simplifying radical expressions helps in condensing your formulas into manageable forms. This makes it easier to perform further calculations or graph the relationships accurately.