- Combine the two radicals: \(\sqrt[3]{16 x^7} \cdot \sqrt[3]{12 x^9} = \sqrt[3]{192 x^{16}}\).
- Factorize the radicand to find perfect cubes: \(192 = 64 \cdot 3 = 4^3 \cdot 3\) and \(x^{16} = x^{15} \cdot x = (x^5)^3 \cdot x\).
- Simplify the cube root: \(\sqrt[3]{4^3 \cdot x^{15} \cdot 3 \cdot x} = 4 x^5 \sqrt[3]{3 x}\).
- The simplified result is \(\boxed{4 x^5(\sqrt[3]{3 x})}\).
Explanation
- Analyzing the Expression
The problem asks to simplify the product of two cube roots: \(\sqrt[3]{16 x^7}\) and \(\sqrt[3]{12 x^9}\). Using the property of radicals that \(\sqrt[n]{A} \cdot \sqrt[n]{B} = \sqrt[n]{A \cdot B}\), we can combine these into a single expression to simplify them effectively. The goal is to extract any perfect cube factors from the resulting radicand.
- Combining the Radicals
Combining the terms under a single cube root, we have:
\[\sqrt[3]{16 x^7} \cdot \sqrt[3]{12 x^9} = \sqrt[3]{(16 \cdot 12) \cdot (x^7 \cdot x^9)}\]
First, calculating the numerical part: \(16 \cdot 12 = 192\).
Next, calculating the variable part: \(x^7 \cdot x^9 = x^{7+9} = x^{16}\).
So, the expression becomes:
\[\sqrt[3]{192 x^{16}}\]
- Simplifying the Radicand
To simplify \(\sqrt[3]{192 x^{16}}\), we look for the largest perfect cube factors within the radicand.
For the numerical part, we know that \(64\) is a perfect cube (\(4^3 = 64\)). Since \(192 = 64 \cdot 3\), we can write \(\sqrt[3]{192} = \sqrt[3]{64 \cdot 3} = 4 \sqrt[3]{3}\).
For the variable part, we look for the largest power of \(x\) that is a multiple of \(3\). We can write \(x^{16} = x^{15} \cdot x^1 = (x^5)^3 \cdot x\).
Thus, \(\sqrt[3]{x^{16}} = \sqrt[3]{(x^5)^3 \cdot x} = x^5 \sqrt[3]{x}\).
- Conclusion and Final Result
Combining these simplified parts together, we get:
\[\sqrt[3]{192 x^{16}} = \sqrt[3]{64 \cdot 3 \cdot x^{15} \cdot x} = \sqrt[3]{64 \cdot x^{15}} \cdot \sqrt[3]{3x} = 4 \cdot x^5 \cdot \sqrt[3]{3x}\]
Comparing this result with the given options, we see that it matches the option \(4 x^5(\sqrt[3]{3 x})\).
Examples
In data science and engineering, simplifying complex radical expressions is often necessary when working with geometric scaling factors or signal processing. For instance, if you are calculating the volume of a sphere based on an expression containing cube roots of variable measurements, simplifying these expressions allows you to identify the relationship between the final volume and the input variables, making it easier to analyze the sensitivity of your model to changes in the dimensions.