What is the simplified form of ( \sqrt{2}+\sqrt{8} ) on the number line?

Correct answer: (3\sqrt{2})

The direct answer is option A: \(3\sqrt2\). Simplify the perfect-square factor inside \(\sqrt8\): since \(8=4\times2\), \(\sqrt8=\sqrt4\sqrt2=2\sqrt2\). Thus \(\sqrt2+\sqrt8=\sqrt2+2\sqrt2=3\sqrt2\). These are like radicals because both contain \(\sqrt2\), so their coefficients 1 and 2 can be added. Option A is correct. Option B, \(\sqrt{10}\), incorrectly combines the numbers under separate square roots; in general \(\sqrt a+\sqrt b\ne\sqrt{a+b}\). Option C, \(2\sqrt{10}\), makes the same invalid combination and adds an extra factor. Option D, \(4\sqrt2\), treats \(\sqrt8\) as \(3\sqrt2\) instead of \(2\sqrt2\). A useful check is that \(\sqrt2\approx1.414\), so the sum is about 4.242, matching \(3\sqrt2\). Memory cue: simplify each radical first, then combine only like radicals.