Express \( 192^{\frac{1}{5}} \) in simplest radical form.

To express \( 192^{\frac{1}{5}} \) in simplest radical form, we first factor 192 into its prime factors. Starting with 192, we can divide by 2 repeatedly: - \( 192 \div 2 = 96 \) - \( 96 \div 2 = 48 \) - \( 48 \div 2 = 24 \) - \( 24 \div 2 = 12 \) - \( 12 \div 2 = 6 \) - \( 6 \div 2 = 3 \) Thus, we have: \[ 192 = 2^6 \times 3^1 \] Now we can express \( 192^{\frac{1}{5}} \) using these prime factors: \[ 192^{\frac{1}{5}} = (2^6 \cdot 3^1)^{\frac{1}{5}} = 2^{\frac{6}{5}} \cdot 3^{\frac{1}{5}} \] To simplify \( 2^{\frac{6}{5}} \): \[ 2^{\frac{6}{5}} = 2^{1 + \frac{1}{5}} = 2^1 \cdot 2^{\frac{1}{5}} = 2 \cdot 2^{\frac{1}{5}} \] Combining everything together, we get: \[ 192^{\frac{1}{5}} = 2 \cdot 2^{\frac{1}{5}} \cdot 3^{\frac{1}{5}} = 2 \cdot (2 \cdot 3)^{\frac{1}{5}} = 2 \cdot 6^{\frac{1}{5}} \] Now we can express this in radical form: \[ 192^{\frac{1}{5}} = 2 \sqrt[5]{6} \] In conclusion, \( 192^{\frac{1}{5}} \) in simplest radical form is: \[ \boxed{2 \sqrt[5]{6}} \]