Go step by step to reduce the radical.
square root of, 224, end square root
224
square root of, 4, end square root, dot, square root of, 56, end square root
4
⋅
56
2, square root of, 56, end square root2
56
2, square root of, 4, end square root, dot, square root of, 14, end square root2
4
⋅
14
2, dot, 2, square root of, 14, end square root2⋅2
14
square root of, empty space, end square root
x
Multiply the two numbers in front of the radical.
Keep the inside of the radical the same.
In this case, we have the square root of 224, which can be simplified as follows.
Step 1: Break down the number 224 into its prime factors.
224 = 2^5 * 7
Step 2: Group the prime factors in pairs. Since we are taking the square root, each pair of the same prime factor can come out of the square root.
square root of 224 = square root of (2^4 * 2 * 7)
Step 3: Simplify the square root of each pair.
square root of (2^4 * 2 * 7) = 2^2 * square root of (2 * 7)
Step 4: Multiply the numbers outside the square root.
2^2 = 4
Step 5: Simplify the remaining radical.
square root of (2 * 7) = square root of 14
Step 6: Put everything together.
square root of 224 = 4 * square root of 14
To reduce the radical √224, we can start by finding the factors of 224.
1. Divide 224 by 2: 224 ÷ 2 = 112.
2. Divide 112 by 2: 112 ÷ 2 = 56.
3. Divide 56 by 2: 56 ÷ 2 = 28.
4. Divide 28 by 2: 28 ÷ 2 = 14.
Now we can rewrite the radical as √(2 × 2 × 2 × 7).
Simplifying inside the radical, we have √(4 × 14).
Taking out the square root of 4, we get 2. The rest of the radical remains the same.
So, the simplified form of √224 is 2√14.
To reduce the radical √224, we need to break it down step by step.
First, we can look for perfect square factors of 224. The perfect squares between 1 and 224 are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, and 196. We can see that 224 is not a perfect square, so none of these numbers can be taken out of the square root.
Next, we can look for factors of 224 that are not perfect squares. One such factor is 4. We can rewrite 224 as 4 * 56.
Now, we have the square root of 4 multiplied by the square root of 56 (√4 * √56). The square root of 4 is 2, so we have 2√56.
We can continue to simplify further, if possible. We can look for perfect square factors of 56. The perfect squares between 1 and 56 are 1, 4, 9, 16, 25, and 36. None of these numbers are factors of 56, so we keep the inside of the radical the same.
Therefore, the simplified form of √224 is 2√56.