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4\times 2\sqrt{1111}
Factor 4444=2^{2}\times 1111. Rewrite the square root of the product \sqrt{2^{2}\times 1111} as the product of square roots \sqrt{2^{2}}\sqrt{1111}. Take the square root of 2^{2}.
8\sqrt{1111}
Multiply 4 and 2 to get 8.