- Factor the first radicand: \(375 = 5^3 \times 3\), so \(\sqrt[3]{375x} = 5\sqrt[3]{3x}\).
- Factor the second radicand: \(x^7 = x^6 \times x = (x^2)^3 \times x\), so \(\sqrt[3]{3x^7} = x^2\sqrt[3]{3x}\).
- Simplify the expression: \(8x^2(5\sqrt[3]{3x}) + 2(x^2\sqrt[3]{3x}) = 40x^2\sqrt[3]{3x} + 2x^2\sqrt[3]{3x}\).
- Combine the results: \(42x^2\sqrt[3]{3x}\).
- The final answer is \(\boxed{42x^2\sqrt[3]{3x}}\).
Explanation
- Problem Analysis
We are given the expression \(E = 8x^2\sqrt[3]{375x} + 2\sqrt[3]{3x^7}\) with the constraint that \(x
eq 0\). Our goal is to simplify this expression by factoring the radicands to extract perfect cubes, which will allow us to simplify the cube roots and then combine the like terms.
- Simplifying the First Term
First, we simplify the first term, \(8x^2\sqrt[3]{375x}\). We look for perfect cube factors of 375. Since \(375 = 125 \times 3\) and \(125 = 5^3\), we can write \(\sqrt[3]{375x} = \sqrt[3]{5^3 \times 3 \times x} = 5\sqrt[3]{3x}\). Multiplying this by the coefficient \(8x^2\), we get:
\[8x^2 \times 5\sqrt[3]{3x} = 40x^2\sqrt[3]{3x}\]
- Simplifying the Second Term
Next, we simplify the second term, \(2\sqrt[3]{3x^7}\). We look for perfect cube factors of \(x^7\). Since \(x^7 = x^6 \times x = (x^2)^3 \times x\), we can simplify the cube root as \(\sqrt[3]{3 \times (x^2)^3 \times x} = x^2\sqrt[3]{3x}\). Multiplying this by the coefficient \(2\), we get:
\[2 \times x^2\sqrt[3]{3x} = 2x^2\sqrt[3]{3x}\]
- Combining Terms and Final Conclusion
Now, we combine the simplified terms:
\[40x^2\sqrt[3]{3x} + 2x^2\sqrt[3]{3x} = (40 + 2)x^2\sqrt[3]{3x} = 42x^2\sqrt[3]{3x}\]
Comparing this result with the given options, we see that it matches the first option. Therefore, the simplified expression is \(42x^2\sqrt[3]{3x}\).
Examples
In engineering and physics, radical expressions often arise when calculating dimensions or volumes of complex shapes. For example, if you are designing a specialized component where the volume is defined by cube roots of variables, simplifying these expressions allows for easier integration into software or manufacturing tools. Understanding how to factor radicands efficiently, as shown here, is fundamental to simplifying complex physical models and ensuring precise calculations in design.