Explain how to simplify√56
-You can show me a factor tree
- you can show me a list of perfect squares
- you can write out an explanation
To simplify √56, there are a few methods you can use:
1. Prime Factorization or Factor Tree:
Start by finding the prime factors of 56. Divide 56 by the smallest prime number possible, which is 2:
56 ÷ 2 = 28
Now, divide 28 by 2 again:
28 ÷ 2 = 14
You can continue dividing by 2 until you cannot divide anymore, which will give you the prime factorization of 56 as 2 × 2 × 2 × 7.
Now, group the factors in pairs, taking one factor from each pair and writing it outside the square root:
√(2 × 2 × 2 × 7) = √(2^2 × 2 × 7)
Since 2^2 is equal to the perfect square 4, we can simplify it further:
√(4 × 2 × 7) = 2√(2 × 7)
So, √56 simplifies to 2√14.
2. List of Perfect Squares:
Make a list of perfect squares and see which one is the largest square that divides 56. We observe that 7 is the largest perfect square that divides 56.
√56 = √(7 × 8)
Now, divide 8 by the largest perfect square that can divide it, which is 4:
√(7 × 4 × 2) = 2√(7 × 2)
Therefore, √56 simplifies to 2√14.
3. Explanation without using factorization:
Since 56 is not a perfect square, we need to simplify it by breaking it down into its prime factors. Since 56 has a factor of 2, we can take the square root of 2 as a common factor outside the square root:
√56 = √(2 × 2 × 2 × 7)
Grouping the factors in pairs, we get:
√(2^2 × 2 × 7)
Since 2^2 is equal to 4, we can simplify it further:
2√(2 × 7)
Thus, √56 simplifies to 2√14.