- Factor the coefficients and variables inside the radicals using prime bases: \(512 = 2^9\) and \(32 = 2^5\).
- Simplify the radicals by pulling out powers of 2 and \(x\) that are multiples of 4: \(\sqrt[4]{2^9 x^{13}} = 4x^3\sqrt[4]{2x}\).
- Simplify the second term by similarly extracting factors: \(2\sqrt[4]{32 x^{13}} = 2 \cdot 2x^3 \sqrt[4]{2x} = 4x^3 \sqrt[4]{2x}\).
- Combine the two identical terms to arrive at the final result: \(\boxed{8x^3 \sqrt[4]{2x}}\).
Explanation
- Analyzing the Expression
We are given the expression \(\sqrt[4]{512 x^{13}} + 2 \sqrt[4]{32 x^{13}}\) where \(x > 0\). Our goal is to simplify this expression by using the property of radicals \(\sqrt[n]{a \cdot b} = \sqrt[n]{a} \cdot \sqrt[n]{b}\) and the laws of exponents. We start by splitting each term into numeric and variable components.
- Simplifying the First Term
First, consider the term \(\sqrt[4]{512 x^{13}}\). We know that \(512 = 2^9\) and \(x^{13} = x^{12} \cdot x\). Applying the root property, we have:
\[\sqrt[4]{512 x^{13}} = \sqrt[4]{2^9 \cdot x^{13}} = \sqrt[4]{2^8 \cdot 2^1 \cdot x^{12} \cdot x^1}\]
Extracting the powers divisible by 4:
\[\sqrt[4]{2^8 \cdot x^{12} \cdot 2x} = \sqrt[4]{2^8} \cdot \sqrt[4]{x^{12}} \cdot \sqrt[4]{2x} = 2^{8/4} \cdot x^{12/4} \cdot \sqrt[4]{2x} = 4x^3 \sqrt[4]{2x}\]
- Simplifying the Second Term
Next, consider the term \(2 \sqrt[4]{32 x^{13}}\). Since \(32 = 2^5\):
\[2 \sqrt[4]{32 x^{13}} = 2 \cdot \sqrt[4]{2^5 \cdot x^{13}} = 2 \cdot \sqrt[4]{2^4 \cdot 2^1 \cdot x^{12} \cdot x^1}\]
Extracting the powers divisible by 4:
\[2 \cdot \sqrt[4]{2^4} \cdot \sqrt[4]{x^{12}} \cdot \sqrt[4]{2x} = 2 \cdot 2^1 \cdot x^3 \cdot \sqrt[4]{2x} = 4x^3 \sqrt[4]{2x}\]
- Combining and Finalizing
Finally, add the two simplified terms together:
\[4x^3 \sqrt[4]{2x} + 4x^3 \sqrt[4]{2x} = 8x^3 \sqrt[4]{2x}\]
The expression has been simplified completely.
Examples
This technique of simplifying radicals is essential in physics and engineering, particularly when dealing with formulas involving square roots, cube roots, or higher-order roots of physical quantities. For instance, in calculating the distance of an object moving under certain acceleration models or determining electrical impedance, you often end up with complex-looking radical expressions. Mastering the ability to combine these terms into a simplified form allows for cleaner calculations and a better intuitive understanding of the underlying relationship between variables.