Rewrite the irrational cube root 3√189 as the product of an integer and another irrational cube root. Show your work.

Answer 1

To rewrite the cube root \( \sqrt[3]{189} \) as the product of an integer and another cube root, we can start by factoring the number 189.

1. Factor 189: \[ 189 = 3 \times 63 \] Now, we can factor 63 further: \[ 63 = 3 \times 21 = 3 \times 3 \times 7 = 3^3 \times 7 \] So, we can rewrite 189: \[ 189 = 3^3 \times 7 \]

2. Express the cube root: Now, we can express the cube root of 189: \[ \sqrt[3]{189} = \sqrt[3]{3^3 \times 7} \]

3. Apply the property of cube roots: Using the property of cube roots, we can separate this into the product of two cube roots: \[ \sqrt[3]{189} = \sqrt[3]{3^3} \times \sqrt[3]{7} \]

4. Simplify: Since the cube root of \( 3^3 \) is simply 3, we have: \[ \sqrt[3]{189} = 3 \times \sqrt[3]{7} \]

So, the final expression for \( \sqrt[3]{189} \) as the product of an integer and another irrational cube root is: \[ \sqrt[3]{189} = 3 \sqrt[3]{7} \]