Rewrite the irrational cube root of ^3√189 as the product of an integer and another irrational cube root. Show your work. Summarize by showing only the work needed to explain your answer

Answer 1

To rewrite the irrational cube root of \( \sqrt[3]{189} \) as the product of an integer and another irrational cube root, we first factor \( 189 \).

1. Factor \( 189 \): \[ 189 = 3 \times 63 = 3 \times 3 \times 21 = 3^3 \times 7 = 27 \times 7 \]

2. Rewrite in cube root form: \[ \sqrt[3]{189} = \sqrt[3]{27 \times 7} \]

3. Separate the cube root: \[ \sqrt[3]{189} = \sqrt[3]{27} \times \sqrt[3]{7} \]

4. Evaluate \( \sqrt[3]{27} \): \[ \sqrt[3]{27} = 3 \]

5. Combine the expressions: \[ \sqrt[3]{189} = 3 \times \sqrt[3]{7} \]

Final Answer: \[ \sqrt[3]{189} = 3 \sqrt[3]{7} \]