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If we use a calculator, we obtain the following result:
[tex]\sqrt[]{19}=4.35889894354[/tex]If we round this number to the nearest hundredth, we have that the number would be 4.36.
The answer that shows the best approximation is 4.4 since this number is the nearest to the actual value of the number. The other approximations, 4.9, 4.7, and 5.1 are not so near as the first option.
Therefore, the answer that shows the best approximation of √19 is 4.4 (first option), since that number is very near to the actual value if we round it to the nearest hundredth (4.36).