Rewrite the irrational cube root 189 as the product of an integer and another irrational cube root. Show your work.
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
To rewrite the irrational cube root of 189 as the product of an integer and another irrational cube root, we can use prime factorization.
1. Start by finding the prime factorization of 189.
189 = 3 * 3 * 3 * 7
2. Group the prime factors in sets of three, since we're looking for the cube root.
(3 * 3 * 3) * 7
3. Simplify the cube inside the parentheses.
27 * 7
4. Rewrite 27 as the cube of an integer, and the remaining 7 as an irrational number.
(3^3) * 7
Therefore, the irrational cube root of 189 can be written as:
∛189 = 3∛7
To rewrite the irrational cube root of 189 as the product of an integer and another irrational cube root, we need to find a perfect cube factor of 189.
To do this, we can prime factorize 189:
189 = 3 * 3 * 3 * 7
Since we are looking for a perfect cube factor, let's group the prime factors in sets of three:
189 = (3 * 3 * 3) * 7
The cube root of (3 * 3 * 3) is 3, thus:
189 = 3 * 3 * 3 * 7
Now, we can rewrite this as the product of an integer and another irrational cube root:
189 = 27 * (7^(1/3))
So, the irrational cube root of 189 can be written as the product of the integer 27 and the irrational cube root of 7.