e) \( \left(5 \cdot 4^{\frac{1}{6}}\right)^{3} \)

Calculate the value by following steps: - step0: Calculate: \(\left(5\times 4^{\frac{1}{6}}\right)^{3}\) - step1: Evaluate the power: \(125\times 2\) - step2: Multiply the numbers: \(250\) To simplify the expression \( \left(5 \cdot 4^{\frac{1}{6}}\right)^{3} \), we can follow these steps: 1. **Apply the exponent to both factors inside the parentheses**: \[ \left(5 \cdot 4^{\frac{1}{6}}\right)^{3} = 5^{3} \cdot \left(4^{\frac{1}{6}}\right)^{3} \] 2. **Calculate \(5^{3}\)**: \[ 5^{3} = 125 \] 3. **Calculate \(\left(4^{\frac{1}{6}}\right)^{3}\)** using the power of a power property: \[ \left(4^{\frac{1}{6}}\right)^{3} = 4^{\frac{3}{6}} = 4^{\frac{1}{2}} = \sqrt{4} = 2 \] 4. **Combine the results**: \[ 125 \cdot 2 = 250 \] Thus, the simplified result of the expression \( \left(5 \cdot 4^{\frac{1}{6}}\right)^{3} \) is \( 250 \).