jmc

We use the distance formula: $\sqrt{(3-0{)}^2+(0-4{)}^2}=\sqrt{9+16}=\boxed{5}$.

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We note that the points $(0,4)$, $(3,0)$, and $(0,0)$ form a right triangle with legs of length 3 and 4. This is a Pythagorean triple, so the hypotenuse must have length $\boxed{5}$.