Evaluate \(1011_2 + 122_4\), leaving your answer in base 10.

Answer Details

To add numbers written in different bases, each number must first be converted into a common base, usually base \(10\), before the addition is carried out.

Converting \(1011_2\) to base \(10\):\[1011_2 = (1 \times 2^3) + (0 \times 2^2) + (1 \times 2^1) + (1 \times 2^0) = 8 + 0 + 2 + 1 = 11_{10}\]Converting \(122_4\) to base \(10\):\[122_4 = (1 \times 4^2) + (2 \times 4^1) + (2 \times 4^0) = 16 + 8 + 2 = 26_{10}\]Adding the two base \(10\) values gives:\[11 + 26 = 37\]

A frequent error is adding the digits of the two numbers as if they were already in the same base without converting first; always convert every number to base \(10\) before combining values from different number systems.