19. \( \left(81^{8}\right)^{\frac{1}{4}} \)

To evaluate the expression \( \left(81^{8}\right)^{\frac{1}{4}} \), we can use the properties of exponents. Here's a step-by-step solution: 1. **Understand the Expression:** The expression is \( \left(81^{8}\right)^{\frac{1}{4}} \). 2. **Apply the Power of a Power Rule:** The power of a power rule states that \( (a^{m})^{n} = a^{m \times n} \). Applying this rule: \[ \left(81^{8}\right)^{\frac{1}{4}} = 81^{8 \times \frac{1}{4}} \] 3. **Multiply the Exponents:** \[ 8 \times \frac{1}{4} = 2 \] So, \[ 81^{8 \times \frac{1}{4}} = 81^{2} \] 4. **Calculate \( 81^{2} \):** \[ 81^{2} = 81 \times 81 = 6561 \] 5. **Final Answer:** \[ \left(81^{8}\right)^{\frac{1}{4}} = 6561 \] **Therefore, \( \left(81^{8}\right)^{\frac{1}{4}} = 6561 \).**