Rewrite the irrational cube root ^3√189 as the product of an integer and another irrational cube root. Show your work.

Answer 1

To rewrite the irrational cube root ^3√189 as the product of an integer and another irrational cube root, we need to find a perfect cube factor of 189.

The prime factorization of 189 is:

189 = 3^3 * 7

We can see that 3^3 = 27 is a perfect cube factor of 189. Now, we can rewrite 189 as:

189 = 27 * 7

Taking the cube root of both sides, we get:

^3√189 = ^3√(27 * 7)

Using the property ^3√(a * b) = ^3√(a) * ^3√(b), we can rewrite this as:

^3√189 = ^3√27 * ^3√7

Simplifying further, we find that:

^3√189 = 3 * ^3√7

Therefore, the irrational cube root ^3√189 can be rewritten as the product of an integer (3) and another irrational cube root (^3√7).