Rewrite the irrational cube root root(189, 3) as the product of an integer and another irrational cube root. Show your work. Note: You will find the option to add a cubic root symbol in the Algebra (x) keyboard.
To rewrite the cube root of 189 as the product of an integer and another irrational cube root, we need to find the largest perfect cube that divides evenly into 189.
The factors of 189 are:
1, 3, 7, 9, 21, 27, 63, 189
Among these factors, the largest perfect cube is 27, which is equal to 3³.
Now, let's rewrite the cube root of 189:
∛189 = ∛(27 × 7)
Using the property of cube roots, we can separate the perfect cube root from the remaining factor:
∛189 = ∛27 × ∛7
Since ∛27 = 3, we can substitute this value:
∛189 = 3 × ∛7
Therefore, the cube root of 189 can be written as the product of the integer 3 and the irrational cube root ∛7.
To rewrite the irrational cube root \(\sqrt[3]{189}\) as the product of an integer and another irrational cube root, we need to find a perfect cube divisor of 189.
Let's find the prime factorization of 189:
189 = 3 × 63
= 3 × (3 × 21)
= 3 × 3 × (3 × 7)
= 3^3 × 7
Now, we have a perfect cube divisor of 189, which is \(3^3\). We can rewrite 189 as:
189 = \(3^3\) × 7
Taking the cube root of both sides:
\(\sqrt[3]{189}\) = \(\sqrt[3]{3^3 \cdot 7}\)
Using the property of radicals, we can write this as the product of two cube roots:
\(\sqrt[3]{189}\) = \(\sqrt[3]{3^3} \cdot \sqrt[3]{7}\)
Simplifying the first cube root:
\(\sqrt[3]{3^3}\) = 3
Therefore, the rewritten form is:
\(\sqrt[3]{189}\) = 3 \(\sqrt[3]{7}\)
So, \(\sqrt[3]{189}\) can be expressed as the product of an integer (3) and another irrational cube root (\(\sqrt[3]{7}\)).
To rewrite the irrational cube root root(189, 3) as the product of an integer and another irrational cube root, we first need to factorize 189.
The prime factorization of 189 is:
189 = 3^3 × 7
Now, we can rewrite root(189, 3) as the product of the cube root of the perfect cube part and the cube root of the remaining part:
root(189, 3) = root(3^3, 3) × root(7, 3)
Simplifying,
root(3^3, 3) × root(7, 3) = 3 × root(7, 3)
Therefore, the irrational cube root root(189, 3) can be rewritten as the product of 3 and root(7, 3).