- Factor 686 into \(2 \times 7^3\) and split exponents \(x^4\) and \(y^7\) into perfect cube components: \(x^3 \cdot x\) and \(y^6 \cdot y\).
- Apply the cube root property to extract the perfect cubes \(\sqrt[3]{7^3} = 7\), \(\sqrt[3]{x^3} = x\), and \(\sqrt[3]{y^6} = y^2\).
- Combine the extracted terms outside the radical with the remaining factor inside the cube root: \(7 x y^2 \sqrt[3]{2 x y}\).
- Match the result to the provided option: \(\boxed{7 x y^2 \sqrt[3]{2 x y}}\).
Explanation
- Prime Factorization and Decomposing Variables
To simplify the expression \(\sqrt[3]{686 x^4 y^7}\), we need to extract the perfect cube factors from the radicand. A perfect cube is a term whose exponent is a multiple of 3.
First, let's look at the numerical coefficient 686. We can perform prime factorization to identify any perfect cubes within it:
\[686 = 2 \times 343 = 2 \times 7^3.\]
Since \(7^3 = 343\), we know that 343 is a perfect cube.
Next, we address the variable parts \(x^4\) and \(y^7\). We decompose their exponents into the largest multiples of 3 plus the remainder:
\[x^4 = x^3 \times x^1\]
\[y^7 = y^6 \times y^1\]
Combining these, we have:
\[\sqrt[3]{686 x^4 y^7} = \sqrt[3]{2 \times 7^3 \times x^3 \times x \times y^6 \times y}\]
- Simplifying the Expression
Now, we group all terms that are perfect cubes together:
\[\sqrt[3]{(7^3 \times x^3 \times y^6) \times (2 \times x \times y)}\]
Using the property of radicals \(\sqrt[n]{ab} = \sqrt[n]{a} \sqrt[n]{b}\), we can separate the perfect cubes:
\[\sqrt[3]{7^3 \times x^3 \times y^6} \times \sqrt[3]{2 \times x \times y}\]
Now, simplify the cube root of the perfect cubes:
\[\sqrt[3]{7^3} = 7\]
\[\sqrt[3]{x^3} = x\]
\[\sqrt[3]{y^6} = (y^2)^3 \implies \sqrt[3]{y^6} = y^2\]
Putting it all together, the simplified expression is:
\[7 x y^2 \sqrt[3]{2 x y}\]
- Comparing Results and Conclusion
Comparing our result \(7 x y^2 \sqrt[3]{2 x y}\) with the given options:
- \(7 x y^2 \sqrt[3]{2 x y}\)
- \(7 x y^3 \sqrt[3]{2 x y}\)
- \(7 x y \sqrt[3]{x y^2}\)
- \(7 x^2 y \sqrt[3]{2 x y^2}\)
The simplified expression matches the first option. Therefore, the correct simplification is \(\boxed{7 x y^2 \sqrt[3]{2 x y}}\).
Examples
In fields like physics or engineering, simplifying complex radical expressions is often necessary when working with volume calculations, specifically when dealing with spheres or cubic structures. For example, if you know the volume of a cubic component is given by \(V = 686 x^4 y^7\) cubic units, you would need to calculate the side length \(s = \sqrt[3]{V}\). Simplifying this expression to \(s = 7xy^2\sqrt[3]{2xy}\) makes the value much easier to manage in further calculations, such as determining the surface area or material cost. This process of isolating cube roots is a standard algebraic skill used to maintain precision while working with physical dimensions.