Simplify under root (12+6 root 3) + under root (12- 6 root 3)
√(12+6√3) + √(12-6√3)
12+6√3 = 9 + 6√3 + 3 = (3+√3)^2
so, you have just
3+√3 + 3-√3 = 6
To simplify the expression, we can start by simplifying each square root individually.
For the first square root, let's simplify √(12 + 6√3):
Step 1: Recognize that 12 can be factored as 4 * 3.
Step 2: Rewrite the expression: √(4 * 3 + 6√3).
Step 3: Distribute the square root: √(4) * √(3 + 6√3).
Step 4: Simplify the square root of 4 to get 2: 2√(3 + 6√3).
Step 5: Expand the square root: 2√3 + 2√(6√3).
For the second square root, let's simplify √(12 - 6√3):
Step 1: Recognize that 12 can be factored as 4 * 3.
Step 2: Rewrite the expression: √(4 * 3 - 6√3).
Step 3: Distribute the square root: √(4) * √(3 - 6√3).
Step 4: Simplify the square root of 4 to get 2: 2√(3 - 6√3).
Step 5: Expand the square root: 2√3 - 2√(6√3).
Now, we can add the two simplified expressions together:
(2√3 + 2√(6√3)) + (2√3 - 2√(6√3)).
The terms inside the parentheses will cancel out:
2√3 + 2√3 + 2√(6√3) - 2√(6√3).
Combining like terms, we have:
4√3.
Therefore, the simplified expression is 4√3.
To simplify the expression, we need to combine the two radicals and simplify if possible. Let's break it down step by step:
1. Start with the expression: √(12 + 6√3) + √(12 - 6√3)
2. We can notice that both terms have a common radical (√3). Let's focus on simplifying this common radical first.
3. Simplify the first term: √(12 + 6√3) = √[(2√3)^2 + 2(2√3)]
= √[4(3) + 4√3]
= √(12 + 4√3)
4. Simplify the second term: √(12 - 6√3) = √[(2√3)^2 - 2(2√3)]
= √[4(3) - 4√3]
= √(12 - 4√3)
5. Now we have: √(12 + 4√3) + √(12 - 4√3)
6. Combine the two radicals by adding the like terms: √(12 + 4√3) + √(12 - 4√3)
= 2√3 + 2√3
= 4√3
7. Therefore, the simplified form of the expression is 4√3.