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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To rewrite the irrational cube root ∛189 as the product of an integer and another irrational cube root, we can simplify it.
First, let's find the prime factors of 189:
189 = 3 × 3 × 3 × 7
Let's group the prime factors in sets of three:
189 = (3 × 3 × 3) × (7 × 1)
The cube root of 27 (3 × 3 × 3) can be written as ∛27 = 3.
Therefore, we can rewrite ∛189 as 3∛7.
So, the irrational cube root ∛189 can be written as the product of an integer (3) and another irrational cube root (∛7).
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To rewrite the irrational cube root ∛189 as the product of an integer and another irrational cube root, let's simplify it.
First, let's find the prime factors of 189:
189 = 3 × 3 × 3 × 7
Now, let's group the prime factors in sets of three:
189 = (3 × 7) × (3 × 3)
The cube root of 3 × 7 can be written as ∛(3 × 7) = ∛21.
Therefore, we can rewrite ∛189 as ∛21 × ∛(3 × 3) = ∛21 × 3.
So, the irrational cube root ∛189 can be written as the product of an integer (3) and another irrational cube root (∛21).
To rewrite the irrational cube root ^3√189 as the product of an integer and another irrational cube root, let's first factorize 189.
We start by finding the prime factors of 189:
189 ÷ 3 = 63
63 ÷ 3 = 21
21 ÷ 3 = 7
Therefore, the prime factorization of 189 is 3 × 3 × 3 × 7.
Next, we group the factors into sets of three:
(3 × 3 × 3) × 7
The cube root ^3√189 can be rewritten as the cube root of the product of an integer cube and an irrational cube root:
^3√189 = ^3√(3 × 3 × 3) × 7
Let's simplify the cube root of the integer cube as an integer:
^3√(3 × 3 × 3) = ^3√27 = 3
Now we can rewrite the cube root ^3√189 as the product of an integer and another irrational cube root:
^3√189 = 3 × ^3√7
Therefore, the irrational cube root ^3√189 can be rewritten as the product of the integer 3 and the irrational cube root ^3√7.