Let's analyze each expression step-by-step to determine their equivalence and the appropriate radical forms:
1. Expression 1: \((5ab)^{\frac{3}{2}}\)
- To convert this expression into radical form, recall that \(x^{\frac{m}{n}} = \sqrt[n]{x^m}\).
- Here, \((5ab)^{\frac{3}{2}}\) can be rewritten as \(\sqrt{(5ab)^3}\).
- Since \((5ab)^3 = 5^3 \cdot a^3 \cdot b^3\), the radical form is \(\sqrt{5^3 \cdot a^3 \cdot b^3}\).
- Simplifying, \(\sqrt{5^3 \cdot a^3 \cdot b^3}\) gives \(\sqrt{125 a^3 b^3}\).
2. Expression 2: \(\sqrt[3]{5 a^2 b^2}\)
- This expression is already in its radical form.
- It represents the cube root of \(5 \cdot a^2 \cdot b^2\).
3. Expression 3: \(\sqrt{125 a^3 b^3}\)
- This is another radical expression.
- Here, \(125 = 5^3\), so \(\sqrt{125 a^3 b^3}\) can be expressed as \(\sqrt{5^3 \cdot a^3 \cdot b^3}\).
4. Expression 4: \(\sqrt[3]{25 a^2 b^2}\)
- This expression presents the cube root of \(25 \cdot a^2 \cdot b^2\).
- Note that \(25 = 5^2\), so it can be written as \(\sqrt[3]{5^2 \cdot a^2 \cdot b^2}\).
5. Expression 5: \(\sqrt{5 a^3 b^3}\)
- This is a radical expression involving the square root of \(5 \cdot a^3 \cdot b^3\).
- It simplifies to express the radical directly without higher powers of constants.
Now, let's determine if these expressions match or simplify to each other, particularly with \((5ab)^{\frac{3}{2}}\):
- From the step-by-step process, \((5ab)^{\frac{3}{2}}\) simplifies to \(\sqrt{125 a^3 b^3}\).
Given this, the radical form of \((5ab)^{\frac{3}{2}}\) is indeed best represented by:
[tex]\(\sqrt{125 a^3 b^3}\)[/tex]
