To find the square root of 56 using prime factorization, we first need to factor 56 into its prime components.
- Start by dividing 56 by the smallest prime number, which is 2: \[ 56 \div 2 = 28 \]
- Continue factoring 28: \[ 28 \div 2 = 14 \]
- Factor 14: \[ 14 \div 2 = 7 \]
- Since 7 is a prime number, we stop here.
Now, we can express 56 as a product of its prime factors: \[ 56 = 2^3 \times 7^1 \]
Next, we find the square root: \[ \sqrt{56} = \sqrt{2^3 \times 7^1} \]
Using the property of square roots: \[ \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \]
We can break this down: \[ \sqrt{56} = \sqrt{2^2 \times 2^1 \times 7^1} = \sqrt{(2^2)} \times \sqrt{(2^1)} \times \sqrt{(7^1)} \]
Calculating separately: \[ \sqrt{2^2} = 2, \] \[ \sqrt{2^1} = \sqrt{2}, \] \[ \sqrt{7^1} = \sqrt{7} \]
Putting this all together: \[ \sqrt{56} = 2 \times \sqrt{2} \times \sqrt{7} = 2\sqrt{14} \]
Thus, the square root of 56 using prime factorization is: \[ \sqrt{56} = 2\sqrt{14} \]