Question

Rewrite the irrational cube root 189−−−√ as the product of an integer and another irrational cube root. Show your work in sentence form.

Answer 1

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

We can start by dividing 189 by the smallest prime number:

1. Dividing 189 by 3, we have:
\[
189 \div 3 = 63
\]
2. Next, we can factor 63 by continuing to divide by 3:
\[
63 \div 3 = 21
\]
3. Lastly, we divide 21:
\[
21 \div 3 = 7
\]

Thus, the complete factorization of 189 is:
\[
189 = 3^3 \times 7
\]

Now we can express \( \sqrt[3]{189} \) in terms of its prime factors:
\[
\sqrt[3]{189} = \sqrt[3]{3^3 \times 7}
\]

Using the property of cube roots that \( \sqrt[3]{a \times b} = \sqrt[3]{a} \times \sqrt[3]{b} \), we can separate the factors:
\[
\sqrt[3]{189} = \sqrt[3]{3^3} \times \sqrt[3]{7}
\]

Calculating \( \sqrt[3]{3^3} \) gives us:
\[
\sqrt[3]{3^3} = 3
\]

Therefore, we can rewrite \( \sqrt[3]{189} \) as:
\[
\sqrt[3]{189} = 3 \times \sqrt[3]{7}
\]

In conclusion, \( \sqrt[3]{189} \) can be rewritten as the product of an integer and another irrational cube root:
\[
\sqrt[3]{189} = 3 \sqrt[3]{7}
\]