Find the cube root of the following using prime factorization. Leave your answer in index notation. (a) \( 2^{\wedge} 3 \times 5^{\wedge} 6 \) (b) \( 3^{\wedge} 3 \times 5^{\wedge} 9 \times 7^{\wedge} 6 \)

To find the cube roots of the given expressions using prime factorization and express the answers in index notation, follow these steps: ### (a) \( 2^{3} \times 5^{6} \) 1. **Apply the cube root to each factor:** \[ \sqrt[3]{2^{3} \times 5^{6}} = \sqrt[3]{2^{3}} \times \sqrt[3]{5^{6}} \] 2. **Simplify each cube root:** \[ \sqrt[3]{2^{3}} = 2^{3/3} = 2^{1} = 2 \] \[ \sqrt[3]{5^{6}} = 5^{6/3} = 5^{2} \] 3. **Combine the simplified terms:** \[ 2 \times 5^{2} \] **Final Answer for (a):** \[ 2 \times 5^{2} \] --- ### (b) \( 3^{3} \times 5^{9} \times 7^{6} \) 1. **Apply the cube root to each factor:** \[ \sqrt[3]{3^{3} \times 5^{9} \times 7^{6}} = \sqrt[3]{3^{3}} \times \sqrt[3]{5^{9}} \times \sqrt[3]{7^{6}} \] 2. **Simplify each cube root:** \[ \sqrt[3]{3^{3}} = 3^{3/3} = 3^{1} = 3 \] \[ \sqrt[3]{5^{9}} = 5^{9/3} = 5^{3} \] \[ \sqrt[3]{7^{6}} = 7^{6/3} = 7^{2} \] 3. **Combine the simplified terms:** \[ 3 \times 5^{3} \times 7^{2} \] **Final Answer for (b):** \[ 3 \times 5^{3} \times 7^{2} \]