go step by step to reduce the radical
√112
112 = 16 ∙ 7
√112 = √ (16 ∙ 7 ) = √16 ∙ √7 = 4 √7
To reduce the radical √112, follow these steps:
Step 1: Determine the prime factorization of 112.
- Divide 112 by its smallest prime factor, which is 2. The result is 56, so write down 2.
- Divide 56 by its smallest prime factor, which is 2. The result is 28, so write down another 2.
- Divide 28 by its smallest prime factor, which is 2. The result is 14, so write down one more 2.
- Divide 14 by its smallest prime factor, which is 2. The result is 7, which is a prime number, so write down 7.
- Therefore, the prime factorization of 112 is 2 * 2 * 2 * 7.
Step 2: Group the prime factors.
- Group the prime factors in pairs: (2 * 2) * 2 * 7.
- Notice that the numbers inside the parentheses are equal. This means you can take one of those pairs out of the square root.
Step 3: Simplify the radical.
- Take out one pair from the square root: 2 * √(2 * 7).
- Multiply the pair outside the square root: 2 * √14.
Therefore, √112 simplifies to 2√14.