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

Answer 1

To rewrite the irrational cube root 189 as the product of an integer and another irrational cube root, we need to find a perfect cube that can be factored out of 189.

Prime factorizing 189, we have:
189 = 3^3 * 7^1

We can factor out the perfect cube 27 (3^3) from 189:
189 = 27 * 7^1

Now, we can write the original irrational cube root 189 as the product of the cube root of the perfect cube and the remaining root:
∛189 = ∛(27 * 7^1)

Using the property of roots, we separate the factors under the cube root:
∛189 = ∛27 * ∛(7^1)

Simplifying:

∛189 = ∛27 * ∛7

The cube root of 27 is 3, an integer, and the cube root of 7 is an irrational number. Therefore,

∛189 = 3 * ∛7