Rewrite the irrational cube root 189−−−√3 as the product of an integer and another irrational cube root. Show your work.
To simplify the cube root of 189, we can break down 189 into its prime factorization:
189 = 3 × 3 × 3 × 7
Since the cube root of 189 involves √3, we can rewrite it as:
∛189 = ∛(3 × 3 × 3 × 7) = ∛(3^3 × 7)
We can take out the cube root of 3^3 since it is a perfect cube, and it simplifies to 3:
∛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 irrational cube root ∛189 as the product of an integer and another irrational cube root, we need to find the largest perfect cube factor of 189.
To do this, let's break down 189 into prime factors:
189 = 3 * 3 * 3 * 7
Notice that 189 can be expressed as (3^3) * 7, which means that 189 is a perfect cube times 7.
Now, let's rewrite 189∛3 as the product of an integer and another irrational cube root:
∛189 = ∛(3^3 * 7) = ∛(3^3) * ∛7 = 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 irrational cube root ∛189 as the product of an integer and another irrational cube root, we can start by finding the prime factorization of 189.
The prime factorization of 189 can be obtained as follows:
- First, we divide 189 by the smallest prime number, 2. It is not divisible by 2.
- Next, we divide 189 by the next smallest prime number, 3. It is divisible by 3. So, we get 189 ÷ 3 = 63.
- 63 is also divisible by 3. We divide it by 3 again: 63 ÷ 3 = 21.
- The prime factorization of 21 is 3 × 7.
So, the prime factorization of 189 is 3 × 3 × 7.
Now, let's rewrite the cube root of 189 (∛189) as the product of an integer and another cube root.
We can rewrite ∛189 as ∛(3 × 3 × 7). Since 3 × 3 = 9, we can simplify it further as ∛(9 × 7).
Using the property of cube roots that says ∛(a × b) = ∛a × ∛b, we can express ∛(9 × 7) as ∛9 × ∛7.
Therefore, the irrational cube root ∛189 can be rewritten as ∛9 × ∛7.
To summarize, ∛189 = ∛9 × ∛7.
Note: In this explanation, we used prime factorization to identify the factors of 189 and then applied the property of cube roots to rewrite it as the product of an integer (∛9) and another irrational cube root (∛7).