- Factor each radical into perfect cube components: \(40 = 2^3 \cdot 5\), \(625 = 5^3 \cdot 5\), and \(x^{13} = (x^4)^3 \cdot x\).
- Simplify the first radical to \(2x^4 \sqrt[3]{5x}\).
- Simplify the second radical part to \(5x^4 \sqrt[3]{5x}\), which when multiplied by the coefficient 3 results in \(15x^4 \sqrt[3]{5x}\).
- Combine the two terms to get the final result: \(\boxed{17x^4 \sqrt[3]{5x}}\).
Explanation
- Analyzing the Expression
To simplify the expression \(\sqrt[3]{40 x^{13}} + 3 \sqrt[3]{625 x^{13}}\) with \(x > 0\), we begin by breaking down each radical into its prime factors and identifying perfect cube components.
Recall the properties of radicals: \(\sqrt[3]{a \cdot b} = \sqrt[3]{a} \cdot \sqrt[3]{b}\) and \(\sqrt[3]{x^{3n}} = x^n\). For the powers of \(x\), we note that \(x^{13} = x^{12} \cdot x = (x^4)^3 \cdot x\).
- Simplifying the First Term
Next, we simplify the first term \(\sqrt[3]{40 x^{13}}\).
Factoring 40 gives \(40 = 8 \cdot 5 = 2^3 \cdot 5\).
Thus, the first term is:
\[\sqrt[3]{2^3 \cdot 5 \cdot (x^4)^3 \cdot x} = \sqrt[3]{2^3} \cdot \sqrt[3]{(x^4)^3} \cdot \sqrt[3]{5x} = 2 \cdot x^4 \cdot \sqrt[3]{5x} = 2x^4 \sqrt[3]{5x}.\]
- Simplifying the Second Term
Now, we simplify the second term \(3 \sqrt[3]{625 x^{13}}\).
Factoring 625 gives \(625 = 125 \cdot 5 = 5^3 \cdot 5\).
So, the radical part is:
\[\sqrt[3]{5^3 \cdot 5 \cdot (x^4)^3 \cdot x} = \sqrt[3]{5^3} \cdot \sqrt[3]{(x^4)^3} \cdot \sqrt[3]{5x} = 5 \cdot x^4 \cdot \sqrt[3]{5x} = 5x^4 \sqrt[3]{5x}.\]
Multiplying this by the coefficient 3, we get:
\[3 \cdot 5x^4 \sqrt[3]{5x} = 15x^4 \sqrt[3]{5x}.\]
- Final Combination and Conclusion
Finally, we add the two simplified terms together:
\[2x^4 \sqrt[3]{5x} + 15x^4 \sqrt[3]{5x} = (2 + 15)x^4 \sqrt[3]{5x} = 17x^4 \sqrt[3]{5x}.\]
Thus, the simplified form of the expression is \(\boxed{17x^4 \sqrt[3]{5x}}\).
Examples
In engineering, simplifying radical expressions often arises when calculating volumes or dimensions of objects that scale with a power. For instance, if you are designing a tank where volume is proportional to \(\sqrt[3]{x^{13}}\), simplifying your formula can significantly speed up your computations and make them less prone to error when working with different dimensions of \(x\). This algebraic technique is foundational in physics and structural design where efficient formulas are essential for quick analysis.