- Recognize that the expression involves cube roots of variables \(x\) and \(y\).
- Simplify the second term \(\sqrt[3]{x^5 y}\) by extracting the perfect cube \(x^3\), resulting in \(x \sqrt[3]{x^2 y}\).
- Add the two terms \(5x \sqrt[3]{x^2 y}\) and \(2x \sqrt[3]{x^2 y}\) to obtain the simplified expression.
- The final simplified expression is \(\boxed{7 x\left(\sqrt[3]{x^2 y}\right)}\).
Explanation
- Analyzing the Expression
We are given the expression \(5x\sqrt[3]{x^2 y} + 2\sqrt[3]{x^5 y}\) to simplify. Our goal is to manipulate the terms so that they have a common factor, allowing us to combine them. We should focus on the properties of radicals and exponents, specifically that \(\sqrt[n]{ab} = \sqrt[n]{a}\sqrt[n]{b}\) and \(\sqrt[n]{a^n} = a\) (assuming \(a \ge 0\)).
- Simplifying the Second Term
Let's look at the second term: \(2\sqrt[3]{x^5 y}\). We can rewrite the radicand \(x^5 y\) as a product involving a perfect cube:
\[x^5 y = x^3 \cdot x^2 \cdot y\]
Applying the property of radicals \(\sqrt[n]{abc} = \sqrt[n]{a}\sqrt[n]{b}\sqrt[n]{c}\), we get:
\[\sqrt[3]{x^5 y} = \sqrt[3]{x^3} \cdot \sqrt[3]{x^2 y}\]
Since \(\sqrt[3]{x^3} = x\), this simplifies to:
\[\sqrt[3]{x^5 y} = x \sqrt[3]{x^2 y}\]
Now, substituting this back into the original second term, we have:
\[2\sqrt[3]{x^5 y} = 2(x \sqrt[3]{x^2 y}) = 2x \sqrt[3]{x^2 y}\]
- Combining Like Terms and Final Conclusion
Now we substitute the simplified version of the second term back into the original expression:
\[5x\sqrt[3]{x^2 y} + 2x\sqrt[3]{x^2 y}\]
Both terms now share the common factor \(\sqrt[3]{x^2 y}\). We can factor this out:
\[(5x + 2x) \sqrt[3]{x^2 y} = 7x \sqrt[3]{x^2 y}\]
Comparing this result to the provided options, we see that it matches the fourth option.
Examples
In engineering and physics, radical expressions often arise when calculating volumes or using formulas involving powers and roots. For instance, if you are determining the total volume of two geometric structures where one dimension grows with the cube root of the product of variables, simplifying the expression becomes essential to quickly assess the total size or efficiency of the design. Being able to combine similar terms allows you to simplify complex physical models into manageable, readable forms, saving time in calculations and analysis.