To find the square root of 56 using prime factorization, we first need to factor 56 into its prime components.
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Start by dividing 56 by the smallest prime number, which is 2: \[ 56 \div 2 = 28 \]
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Next, divide 28 by 2 again: \[ 28 \div 2 = 14 \]
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Divide 14 by 2 once more: \[ 14 \div 2 = 7 \]
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Finally, 7 is a prime number.
Now we can write the prime factorization of 56: \[ 56 = 2^3 \times 7 \]
To find the square root, we take the square root of the prime factorization: \[ \sqrt{56} = \sqrt{2^3 \times 7} \]
We can separate this into two parts: \[ \sqrt{56} = \sqrt{2^2 \times 2^1 \times 7} = \sqrt{2^2} \times \sqrt{2^1} \times \sqrt{7} \]
Now, calculating each part: \[ \sqrt{2^2} = 2 \quad \text{and} \quad \sqrt{2^1} = \sqrt{2} \] Thus, \[ \sqrt{56} = 2 \times \sqrt{2} \times \sqrt{7} \]
Combining the radicals, we have: \[ \sqrt{56} = 2 \times \sqrt{14} \]
Therefore, the square root of 56 expressed with prime factorization is: \[ \sqrt{56} = 2\sqrt{14} \]