To determine which of the given radical expressions has an absolute value symbol in its simplified form, let's simplify each expression one by one.
-
\(\sqrt[4]{81x}\):
- We can simplify \(81\) as \(3^4\), so \(\root(4)(81) = \root(4)(3^4) = 3\).
- Thus, \(\sqrt[4]{81x} = 3\sqrt[4]{x}\), which does not involve absolute values.
-
\(\sqrt[4]{16x^4}\):
- \(16\) is \(2^4\) and \(x^4\) under the fourth root gives \(\root(4)(16x^4) = \root(4)(16) \cdot \root(4)(x^4) = 2x\).
- Therefore, \(\sqrt[4]{16x^4} = 2x\), which does not involve absolute values.
-
\(\sqrt[3]{64x^3}\):
- \(64\) is \(4^3\), so \(\sqrt[3]{64} = 4\) and \(\sqrt[3]{x^3} = x\).
- Hence, \(\sqrt[3]{64x^3} = 4x\), which does not involve absolute values.
-
\(\sqrt[3]{-125x^3}\):
- \(-125\) can be expressed as \(-5^3\), so \(\sqrt[3]{-125} = -5\) and \(\sqrt[3]{x^3} = x\).
- Therefore, \(\sqrt[3]{-125x^3} = -5x\), which does not involve absolute values.
Now, we check which expressions include an absolute value during simplification. The general rule is that when evaluating an even root of a variable (e.g., \(x\)), it would be necessary to include an absolute value if we do not know the sign of \(x\).
From the calculation, all expressions simplified to forms without absolute value signs, but if \(x\) was being taken to an even root (like the fourth root), then we would have to consider the possibility of \(x\) being negative. In this case, we see:
- The first and second roots do not generate absolute value.
- The third root simplifies nicely to \(4x\) (no absolute value).
- The fourth expression, \(\sqrt[3]{-125x^3}\), does not add an absolute value since the cube root of \(x^3\) just gives \(x\) without needing it.
None of these radical expressions includes an absolute value in their simplified forms based on straightforward calculation, unless the context or constraints indicate otherwise for \(x\).
Therefore, none of the given radical expressions has an absolute value symbol in its simplified form based on the results of their simplifications.