- Factor the coefficient \(56\) into \(2^3 \cdot 7\).
- Break down the variables \(x^7\) and \(y^5\) into powers of 3, yielding \(x^6 \cdot x\) and \(y^3 \cdot y^2\).
- Extract the perfect cubes \(\sqrt[3]{2^3}\), \(\sqrt[3]{x^6}\), and \(\sqrt[3]{y^3}\) to obtain \(2x^2y\).
- The final simplified expression is \(\boxed{2 x^2 y \sqrt[3]{7 x y^2}}\).
Explanation
- Problem Analysis
The expression to simplify is \(\sqrt[3]{56 x^7 y^5}\), where \(x
eq 0\) and \(y
eq 0\). Our goal is to extract perfect cubes from the cube root. A perfect cube is any term whose exponent is a multiple of 3. We can represent the expression as a product of perfect cube components and the remaining parts.
- Factoring the Radicand
First, we factor the numerical coefficient 56. We look for the largest perfect cube that divides 56:
\[56 = 8 \times 7 = 2^3 \times 7.\]
Next, we decompose the variable powers \(x^7\) and \(y^5\) into powers that are multiples of 3 and their remainders:
\[x^7 = x^6 \cdot x^1 = (x^2)^3 \cdot x\]
\[y^5 = y^3 \cdot y^2\]
Substituting these back into the original expression gives:
\[\sqrt[3]{(2^3 \cdot 7) \cdot (x^6 \cdot x) \cdot (y^3 \cdot y^2)}.\]
- Simplifying the Expression
Now, group the perfect cube terms together:
\[\sqrt[3]{(2^3 \cdot x^6 \cdot y^3) \cdot (7 \cdot x \cdot y^2)}.\]
Using the property of cube roots \(\sqrt[3]{a \cdot b} = \sqrt[3]{a} \cdot \sqrt[3]{b}\), we can separate the expression:
\[\sqrt[3]{2^3 \cdot x^6 \cdot y^3} \cdot \sqrt[3]{7 \cdot x \cdot y^2}.\]
Evaluating the first part:
\[\sqrt[3]{2^3} \cdot \sqrt[3]{(x^2)^3} \cdot \sqrt[3]{y^3} = 2 \cdot x^2 \cdot y.\]
Combining this with the remaining cube root gives the simplified expression:
\[2 x^2 y \sqrt[3]{7 x y^2}.\]
- Conclusion
Comparing our simplified result \(2 x^2 y \sqrt[3]{7 x y^2}\) with the given options, we identify that it matches the first choice provided. Thus, the correct expression is \(2 x^2 y \sqrt[3]{7 x y^2}\). Therefore, the final answer is the first option.
Examples
Simplifying radical expressions is a fundamental skill in engineering and physics, often used when dealing with volume formulas where exponents are not perfectly divisible by 3. For example, if you are calculating the side length of a cube with volume \(V = 56 x^7 y^5 \text{ cm}^3\), the side length would be \(s = \sqrt[3]{56 x^7 y^5}\), which simplifies to \(2 x^2 y \sqrt[3]{7 x y^2} \text{ cm}\). Mastering this helps in quickly estimating dimensions and simplifying complex physical models.