Final answer:
The product of the square roots √12, √18, and √30 is simplified to 36√5 by applying the properties of square roots and then factoring the resulting number to simplify the square root.
Step-by-step explanation:
The student is asking to find the product of three square roots: √12, √18, and √30. To simplify this expression, we can use the property of square roots that states √(a) × √(b) = √(a × b). Applying this property step by step we get:
√12 × √18 = √(12 × 18) = √(216),
Now we multiply this result by √30:
√(216) × √30 = √(216 × 30) = √(6480).
To further simplify, we need to factor 6480 into its prime factors and look for pairs of factors to eliminate the square root:
6480 = 2^4 × 3^4 × 5^1
Pull out the pairs from under the square root:
√6480 = √(2^4) × √(3^4) × √5 = 2^2 × 3^2 × √5 = 4 × 9 × √5 = 36√5
So the product of √12, √18, and √30 is 36√5.