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I've never heard of converting an 8 bit binary number to Hex by splitting it to nibbles first.
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One hex digit represents four binary digits, which is why some learn this method. It's actually much easier this way.
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First to answer your question, yes, 0110 would be six and 0011 would be 3, which would make your whole byte as 9.
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Intuitively, it seems like you should add the 6 and the 3, but that would yield the wrong answer, as you pointed out. Instead, using this method, you just concatenate the 6 and the 3. Thus, the answer is 63h, which is the same result you calculated.