Simplify \\(\frac{6+8i}{2}\\).

Find the conjugate needed to simplify \\(\frac{1}{3-2i}\\).

Simplify \\(\frac{1}{1+i}\\).

Simplify \\(\frac{4}{2-i}\\).

Simplify \\(\frac{3+i}{1-i}\\).

Simplify \\(\frac{2-5i}{3+i}\\).

Find the reciprocal of \\(2+3i\\).

Simplify \\(\frac{-1+4i}{2+2i}\\).

Use the formula to simplify \\(\frac{5+2i}{1+3i}\\).

Simplify \\(\frac{(1+i)^2}{1-i}\\).

Verify that \\(\frac{1}{2-i}=\frac25+\frac15i\\).

Explain why the denominator becomes real when dividing by \\(c+di\\) using its conjugate.

Find \\(z\\) if \\((2-i)z=5+3i\\).

Find real \\(a\\) and \\(b\\) if \\(\frac{a+bi}{1+i}=2-i\\).

For which real \\(t\\) is \\(\frac{1+ti}{1-i}\\) real?

For which real \\(t\\) is \\(\frac{t+i}{2+i}\\) purely imaginary?

Find all real \\(t\\) for which \\(\frac{2+ti}{t-i}\\) is real, with \\(t\neq0\\) not assumed.

A learner simplifies \\(\frac{1}{3+4i}\\) as \\(\frac13+\frac14i\\). Diagnose the error and find the reciprocal.

Prove that \\(\frac{1}{a+bi}=\frac{a-bi}{a^2+b^2}\\) when \\(a+bi\neq0\\).

Explain why division by \\(0+0i\\) is not allowed, using the conjugate method.