To find the square root of 56 using prime factorization, we first need to factor 56 into its prime components.
- Factor 56:
- We can divide by 2: \(56 \div 2 = 28\)
- Divide by 2 again: \(28 \div 2 = 14\)
- Divide by 2 again: \(14 \div 2 = 7\)
- Now, 7 is a prime number.
So, the prime factorization of 56 is: \[ 56 = 2^3 \times 7^1 \]
- Finding the square root:
To find the square root, we can take half the exponent of each prime factor:
- For \(2^3\), half of 3 is \(1.5\), so we have \(2^{1.5} = 2^{1} \times \sqrt{2}\).
- For \(7^1\), half of 1 is \(0.5\), so we have \(7^{0.5} = \sqrt{7}\).
Thus, the expression for the square root of 56 is: \[ \sqrt{56} = \sqrt{2^3 \times 7^1} = 2^{1.5} \times 7^{0.5} = 2 \sqrt{2} \times \sqrt{7} = 2 \sqrt{14} \]
So, the square root of 56 can be simplified to: \[ \sqrt{56} = 2\sqrt{14} \]
In summary, the square root of 56 is: \[ \sqrt{56} = 2\sqrt{14} \]