To express \( 128^{1/3} \) in simplest radical form, we start by recognizing that \( 128 \) can be factored into its prime factors.
First, let's find the prime factorization of \( 128 \):
\[ 128 = 2^7 \]
Now, we can rewrite \( 128^{1/3} \) as follows:
\[ 128^{1/3} = (2^7)^{1/3} = 2^{7/3} \]
Next, we can separate the exponent:
\[ 2^{7/3} = 2^{2 + 1/3} = 2^2 \cdot 2^{1/3} \]
Simplifying further:
\[ 2^2 = 4 \]
Thus, we have:
\[ 128^{1/3} = 4 \cdot 2^{1/3} \]
In radical form, \( 2^{1/3} \) can be expressed as \( \sqrt[3]{2} \). Therefore:
\[ 128^{1/3} = 4 \sqrt[3]{2} \]
In conclusion, the simplest radical form of \( 128^{1/3} \) is:
\[ \boxed{4\sqrt[3]{2}} \]