To rewrite the irrational cube root \( \sqrt[3]{\frac{3}{189}} \) as the product of an integer and another irrational cube root, we can start by simplifying the fraction inside the cube root.
First, let's simplify \( \frac{3}{189} \):
\[ \frac{3}{189} = \frac{3}{3 \times 63} = \frac{1}{63} \]
Now, we can rewrite the cube root:
\[ \sqrt[3]{\frac{3}{189}} = \sqrt[3]{\frac{1}{63}} \]
Next, we can separate the fraction inside the cube root:
\[ \sqrt[3]{\frac{1}{63}} = \sqrt[3]{\frac{1}{3 \times 21}} = \sqrt[3]{1} \cdot \sqrt[3]{\frac{1}{63}} = 1 \cdot \sqrt[3]{\frac{1}{63}} = \sqrt[3]{\frac{1}{63}} \]
Now, we can express \( \frac{1}{63} \) as \( \frac{1}{3^2 \times 7} \):
\[ \sqrt[3]{\frac{1}{63}} = \sqrt[3]{\frac{1}{3^2 \times 7}} = \frac{1}{\sqrt[3]{3^2 \times 7}} \]
Now we can find the cube root of \( 63 \) as follows:
\[ 63 = 3^2 \cdot 7 \]
Then we can break \( \sqrt[3]{63} \) into its components:
\[ \sqrt[3]{63} = \sqrt[3]{3^2} \cdot \sqrt[3]{7} \]
Now combine the expressions back together, using \( 1 \) from our earlier work:
We can express: \[ \sqrt[3]{\frac{1}{63}} = \frac{1}{\sqrt[3]{63}} \]
Thus, rewriting the expression: \[ \sqrt[3]{\frac{3}{189}} = \sqrt[3]{\frac{1}{63}} = \frac{1}{\sqrt[3]{63}} = \frac{1}{\sqrt[3]{3^2 \cdot 7}} \]
Final result: We multiply to bring the cube root out of the denominator, if needed.
Thus, we can write it finally like: \[ \sqrt[3]{\frac{3}{189}} = \frac{1}{3} \cdot \sqrt[3]{\frac{1}{7}} \]
So we can express \( \sqrt[3]{\frac{3}{189}} \) as:
\[ = \frac{1}{3} \cdot \sqrt[3]{\frac{1}{7}} \]
This is not quite in a simplified cube root format but there’s not a clean integer with a specific irrational cube root unless we revert back to handling 63 and writing it accordingly as \( 1\cdot \sqrt[3]{3/7} \).
Ultimately, the final simplified product is: \[ = \frac{1}{3} \cdot \sqrt[3]{\frac{1}{7}} \] or similarly, as
\[ = \frac{1}{\sqrt[3]{63}} \]
In the context required: \[ \sqrt[3]{\frac{3}{189}} = \frac{1}{3} \cdot \sqrt[3]{7} \] is generally what we would condense down if just simplifying cube roots.
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