Roots and Irrational Numbers.

Question

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

Let's start by finding the prime factorization of 189:

189 = 3 * 63
= 3 * 3 * 21
= 3 * 3 * 3 * 7

From the prime factorization, we can see that 189 is divisible by 3 three times.

Now, let's rewrite the cube root of 189:

^3√189 = ^3√(3 * 3 * 3 * 7)

We can take out one of the perfect cubes:

^3√(3 * 3 * 3 * 7) = ^3√(3^3 * 7)

This can be rewritten as:

^3√(3^3) * ^3√7

So, ^3√189 is equivalent to 3 * ^3√7.