Express the radical in simplied form:
The 5th root of 2430
I do not know how to type in the radical sign.
Look for 5th powers and factor them out:
2430 = 2 * 3^5 * 5
root(2430) = 3*root(10)
To express the radical in simplified form, we can start by finding the prime factorization of the number inside the radical.
The prime factorization of 2430 can be found by dividing it by prime numbers starting from 2. The prime factorization of 2430 is:
2430 = 2 × 3 × 405
= 2 × 3 × 3 × 135
= 2 × 3 × 3 × 3 × 45
= 2 × 3 × 3 × 3 × 3 × 15
= 2 × 3 × 3 × 3 × 3 × 3 × 5
Now, let's group the factors in sets of 5 (since we want to find the 5th root):
(2 × 3 × 3 × 3 × 3 × 3 × 5) = (2 × 3 × 3 × 3 × 3) × (3 × 5)
= (2 × 3^5) × (3 × 5)
Now, we can take out the 5th root of each group:
∛(2 × 3^5) × (3 × 5)
Since the 5th root (∛) of 2 × 3^5 is 3^1 (as 3^1 × 3 × 5 = 3^2 × 5) and the 5th root of (3 × 5) is (3 × 5)^(1/5), the simplified form of the radical is:
3 × (3 × 5)^(1/5)
Therefore, the simplified form of the 5th root of 2430 is 3 × (3 × 5)^(1/5).