- Factorize the terms inside the radicals to reveal perfect fourth powers: \(32=2^4 \cdot 2\) and \(512=2^8 \cdot 2\), and \(x^{14}=x^{12} \cdot x^2\).
- Simplify the first term: \(-4\sqrt[4]{2^4 \cdot 2 \cdot x^{12} \cdot x^2} = -4 \cdot 2 \cdot x^3 \sqrt[4]{2x^2} = -8x^3\sqrt[4]{2x^2}\).
- Simplify the second term: \(-\sqrt[4]{2^8 \cdot 2 \cdot x^{12} \cdot x^2} = -4 \cdot x^3 \sqrt[4]{2x^2} = -4x^3\sqrt[4]{2x^2}\).
- Combine the two results to get the final simplified expression: \(-12x^3\sqrt[4]{2x^2}\).
- The final answer is \(\boxed{-12 x^3 \sqrt[4]{2 x^2}}\).
Explanation
- Analyzing the Expression
We are given the expression \(-4 \sqrt[4]{32 x^{14}} - \sqrt[4]{512 x^{14}}\) with the condition \(x > 0\). Our goal is to simplify this expression by breaking down the constants and variables inside the fourth roots. Recall that for any positive real numbers \(a\) and \(b\), \(\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}\), and for \(x > 0\), \(\sqrt[n]{x^{m}} = x^{m/n}\).
- Simplifying the First Term
First, we break down the numerical components into powers of 2. We have \(32 = 2^5 = 2^4 \cdot 2^1\) and \(512 = 2^9 = 2^8 \cdot 2^1\). For the variable part, we have \(x^{14} = x^{12} \cdot x^2\).
Now, we simplify the first term: \(-4 \sqrt[4]{32 x^{14}} = -4 \sqrt[4]{2^4 \cdot 2 \cdot x^{12} \cdot x^2}\).
Extracting the fourth roots of the perfect powers, we get \(\sqrt[4]{2^4} = 2\) and \(\sqrt[4]{x^{12}} = x^{12/4} = x^3\).
Thus, the first term becomes \(-4 \cdot 2 \cdot x^3 \sqrt[4]{2 x^2} = -8 x^3 \sqrt[4]{2 x^2}\).
- Simplifying the Second Term
Next, we simplify the second term: \(-\sqrt[4]{512 x^{14}} = -\sqrt[4]{2^8 \cdot 2 \cdot x^{12} \cdot x^2}\).
Extracting the fourth roots of the perfect powers, we get \(\sqrt[4]{2^8} = 2^{8/4} = 2^2 = 4\) and \(\sqrt[4]{x^{12}} = x^3\).
Thus, the second term becomes \(-1 \cdot 4 \cdot x^3 \sqrt[4]{2 x^2} = -4 x^3 \sqrt[4]{2 x^2}\).
- Combining Like Terms and Final Result
Finally, we combine the two simplified terms:
\(-8 x^3 \sqrt[4]{2 x^2} - 4 x^3 \sqrt[4]{2 x^2}\).
Since both terms contain the common factor \(x^3 \sqrt[4]{2 x^2}\), we can subtract the coefficients:
\(-8 - 4 = -12\).
Therefore, the fully simplified expression is \(-12 x^3 \sqrt[4]{2 x^2}\).
Examples
Radical simplification is a fundamental skill in algebra that allows engineers and scientists to handle complex formulas efficiently. For instance, in calculating the physical dimensions of objects based on volume or surface area formulas that involve powers and roots, simplifying expressions like \(-4 \sqrt[4]{32 x^{14}} - \sqrt[4]{512 x^{14}}\) into \(-12 x^3 \sqrt[4]{2 x^2}\) helps in reducing computational complexity and makes it easier to evaluate values for specific material parameters. This process ensures that complex mathematical models remain manageable and easier to interpret in real-world applications.