Rewrite the irrational cube root root(189, 3) as the product of an integer and another irrational cube root. Show your work.
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(2 points)
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To rewrite the irrational cube root ∛189 as the product of an integer and another irrational cube root, we can split the prime factorization of 189 into perfect cube factors.
The prime factorization of 189 is 3 * 3 * 3 * 7.
We can rewrite 189 as (3 * 3 * 3 * 7) = (3 * 3) * (3 * 7).
Taking the cube root of 3 * 3, we get ∛(3 * 3) = 3.
The remaining factor is 3 * 7, which is 21. So, the cube root of 21 is ∛21.
Therefore, ∛189 can be rewritten as 3 * ∛21.
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 factor of 189.
We can start by finding the prime factorization of 189:
189 = 3 * 63
= 3 * 3 * 21
= 3 * 3 * 3 * 7
From the prime factorization, we see that the cube root of 189 can be written as
∛189 = ∛(3 * 3 * 3 * 7)
Next, we can group the perfect cube factors together. In this case, the perfect cube factor is 3 * 3 * 3, which is equal to (3³):
∛189 = ∛(3³ * 7)
Now, we can rewrite the cube root of 189 as the product of an integer and another irrational cube root:
∛189 = 3∛7
Therefore, the irrational cube root (∛189) can be rewritten as the product of an integer (3) and another irrational cube root (∛7).
To rewrite the cube root of 189, we need to find a perfect cube factor of 189. By prime factorizing 189, we can identify any perfect cube factors.
The prime factorization of 189 is:
189 = 3 * 63
= 3 * 3 * 21
= 3 * 3 * 3 * 7
Here we can see that 189 has a perfect cube factor of 3 * 3 * 3 = 27.
Therefore, we can rewrite the cube root of 189 as:
cube root of 189 = cube root of (27 * 7)
Using the property of cube roots, we can rewrite it as the product of an integer and another cube root:
cube root of (27 * 7) = cube root of 27 * cube root of 7
Simplifying further, the cube root of 27 is 3, which leaves us with:
cube root of 189 = 3 * cube root of 7
So, the irrational cube root of 189 can be rewritten as the product of an integer (3) and another irrational cube root (cube root of 7).