6. Perform the indicated operation and write your answer in standard form.
\[\left( {2 + 7i} \right)\left( {8 + 3i} \right)\]We know how to multiply two polynomials and so we also know how to multiply two complex numbers. All we need to do is "foil" the two complex numbers to get,
\[\left( {2 + 7i} \right)\left( {8 + 3i} \right) = 16 + 6i + 56i + 21{i^2}\]All we need to do to finish the problem is to recall that \({i^2} = - 1\). Upon using this fact we can finish the problem.
\[\left( {2 + 7i} \right)\left( {8 + 3i} \right) = 16 + 6i + 56i + 21\left( { - 1} \right) = \require{bbox} \bbox[2pt,border:1px solid black]{{ - 5 + 62i}}\]