Question

Rewrite the irrational cube root 3/189 as the product of an integer and another irrational cube root. Show your work.
Note: You will find the option to add a cubic root symbol in the Algebra (X) keyboard.

Answer 1

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.