- Decompose the expression \(\sqrt[4]{144 a^{12} b^3}\) into \(\sqrt[4]{144} \cdot \sqrt[4]{a^{12}} \cdot \sqrt[4]{b^3}\).
- Simplify each part: \(\sqrt[4]{144} = 2\sqrt[4]{9}\), \(\sqrt[4]{a^{12}} = a^3\), and \(\sqrt[4]{b^3}\) remains as is.
- Combine the results to get the equivalent expression \(2 a^3 \sqrt[4]{9 b^3}\).
- The correct choice is the first one, \(\boxed{2 a^3 \sqrt[4]{9 b^3}}\).
Explanation
- Decomposing the Expression
We want to simplify the expression \(\sqrt[4]{144 a^{12} b^3}\) given the constraints \(a \geq 0\) and \(b \geq 0\). The expression is a fourth root, which can be distributed over the factors inside the radicand. Recall that for any non-negative \(x, y,\) and positive \(n, m,\) we have \(\sqrt[n]{x \cdot y} = \sqrt[n]{x} \cdot \sqrt[n]{y}\) and \(\sqrt[n]{x^m} = x^{m/n}\). Applying these rules, the original expression can be broken down into \(\sqrt[4]{144} \cdot \sqrt[4]{a^{12}} \cdot \sqrt[4]{b^3}\).
- Simplifying Components
We simplify each part of the expression individually.
First, for the variable part \(a^{12}\), we have \(\sqrt[4]{a^{12}} = a^{12/4} = a^3\).
Second, for the variable part \(b^3\), we leave it as \(\sqrt[4]{b^3}\).
Finally, for the numerical part 144, we note that \(144 = 12^2\). Therefore, \(\sqrt[4]{144} = \sqrt[4]{12^2} = 12^{2/4} = 12^{1/2} = \sqrt{12}\). Since \(\sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3}\), we can rewrite this as \(\sqrt[4]{3^2} \cdot 2 = 2\sqrt[4]{9}\).
- Combining and Final Comparison
Now we combine the simplified components:
\(\sqrt[4]{144} \cdot \sqrt[4]{a^{12}} \cdot \sqrt[4]{b^3} = (2\sqrt[4]{9}) \cdot (a^3) \cdot (\sqrt[4]{b^3})\).
By multiplying these together, we get:
\(2 a^3 \sqrt[4]{9 \cdot b^3} = 2 a^3 \sqrt[4]{9b^3}\).
Comparing this to the given options, we see that it matches the first choice provided.
Examples
In fields like engineering and physics, radicals are common in formulas for wave propagation or stress distribution. For instance, if you are designing a structural component where the capacity is related to a fourth-root function of material constants (such as stiffness, length, and width), simplifying these expressions allows for faster calculations and more intuitive understanding of how individual variables influence the overall design capacity. Mastering these algebraic simplifications is a foundational skill for analyzing such complex physical relationships efficiently.