- Factor the radicand: \(686 = 2 \cdot 7^3\), \(x^4 = x^3 \cdot x\), and \(y^7 = y^6 \cdot y\).
- Simplify each radical part separately: \(\sqrt[3]{7^3 \cdot 2} = 7\sqrt[3]{2}\), \(\sqrt[3]{x^3 \cdot x} = x\sqrt[3]{x}\), and \(\sqrt[3]{y^6 \cdot y} = y^2\sqrt[3]{y}\).
- Multiply the simplified parts to get \(\boxed{7 x y^2 \sqrt[3]{2 x y}}\).
Explanation
- Analysis of the Expression
We are asked to simplify the expression \(\sqrt[3]{686 x^4 y^7}\). To simplify a cube root, we aim to extract any perfect cube factors from the radicand. The expression can be decomposed into three parts: the constant, the variable \(x\), and the variable \(y\). Specifically, we have \(\sqrt[3]{686} \cdot \sqrt[3]{x^4} \cdot \sqrt[3]{y^7}\).
- Simplifying the Constant Part
First, consider the constant factor 686. The prime factorization of 686 is \(686 = 2 \times 343 = 2 \times 7^3\). Since 343 is a perfect cube (\(7^3 = 343\)), we can simplify \(\sqrt[3]{686}\) as \(\sqrt[3]{7^3 \times 2} = 7 \sqrt[3]{2}\).
- Simplifying the Variables
Next, simplify the variables using the property \(\sqrt[3]{a^n} = a^{\lfloor n/3 \rfloor} \sqrt[3]{a^{n \pmod 3}}\):
For \(x^4\): \(\sqrt[3]{x^4} = \sqrt[3]{x^3 \cdot x} = x \sqrt[3]{x}\).
For \(y^7\): \(\sqrt[3]{y^7} = \sqrt[3]{(y^2)^3 \cdot y} = y^2 \sqrt[3]{y}\).
- Combining Results and Final Conclusion
Now combine the simplified parts together:
\[ 7 \sqrt[3]{2} \times x \sqrt[3]{x} \times y^2 \sqrt[3]{y} = 7 x y^2 \sqrt[3]{2 \cdot x \cdot y} \]
Comparing this result to the given options, we identify that the simplified expression is \(7 x y^2 \sqrt[3]{2 x y}\).
Examples
Simplifying radicals is a fundamental skill in algebra, useful in fields ranging from physics to engineering. For example, when calculating the volume of a sphere or the diagonal of a 3D box, you often end up with complex radicals. Simplifying these expressions allows you to compare values more easily, perform further algebraic manipulations, or interpret the physical meaning of a formula. Mastering this helps you manage complex data efficiently.