To find the cubic root of 189, we can simplify it as follows:
First, the prime factorization of 189: 189 = 3 × 63 63 = 3 × 21 21 = 3 × 7
So, we have: 189 = \(3^3 \times 7\)
Now we can find the cubic root: \[ \sqrt[3]{189} = \sqrt[3]{3^3 \times 7} = \sqrt[3]{3^3} \times \sqrt[3]{7} = 3 \times \sqrt[3]{7} \]
Thus, \(\sqrt[3]{189} = 3 \sqrt[3]{7}\).
Now let's check the given options to see which one matches:
- \(27^\frac{1}{3}\sqrt{7} = 3\sqrt[3]{7}\)
- \(9^\frac{1}{3}\sqrt{21}\)
- \(3\sqrt[3]{7}\)
- \(3^\frac{1}{3}\sqrt{7}\)
The correct answer is \(3\sqrt[3]{7}\).