Rewrite the irrational cube root 3√189 as the product of an integer and another irrational cube root. Show your work.
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We can rewrite 189 as the product of a perfect cube and a factor that is not a perfect cube.
The prime factorization of 189 is 189 = 3 × 63.
Breaking down 63 further, we have 63 = 3 × 21.
Breaking down 21 further, we have 21 = 3 × 7.
Therefore,
189 = 3 × 3 × 3 × 7
We can group the perfect cube factors together:
189 = (3 × 3 × 3) × 7 = 27 × 7
Now, let's rewrite the original expression:
∛189 = ∛(27 × 7)
Using the properties of radicals, we can separate the cube root into a product of two cube roots:
∛189 = ∛27 × ∛7
The cube root of 27 is 3, so we can rewrite it as:
∛189 = 3∛7
Therefore, the irrational cube root ∛189 can be rewritten as the product of an integer 3 and the irrational cube root ∛7.