Multiply \\((2+i)(3+i)\\).

Find \\(i(4-7i)\\).

Multiply \\((5+2i)(1-3i)\\).

Compute \\((-2+3i)(4-i)\\).

Use \\((a+bi)(c+di)=(ac-bd)+(ad+bc)i\\) to multiply \\((3-2i)(5+4i)\\).

Find \\((1+2i)^2\\).

Multiply \\((2-i)(2+i)\\) and explain why the answer is real.

Find \\((-3+2i)^2\\).

Compute \\((1+i)(2-i)(3+i)\\).

Find \\(z^2\\) for \\(z=4-3i\\).

Show that \\(|(3+4i)(1-2i)|=|3+4i|\\,|1-2i|\\).

Explain why multiplying by \\(i\\) sends \\(a+bi\\) to \\(-b+ai\\).

Find real \\(x\\) if \\((x+i)(2-i)=7+0i\\).

Find \\(a\\) and \\(b\\) if \\((a+bi)(1+i)=3+7i\\).

For which real \\(t\\) is \\((t+i)(t-i)=10\\)?

For which real \\(t\\) is \\((1+ti)^2\\) purely imaginary?

Find all real \\(t\\) for which \\((t+2i)(3-ti)\\) is real.

A learner says \\((2+3i)^2=4+9i^2=-5\\). Diagnose the error and correct the product.

Prove that \\((a+bi)(a-bi)=a^2+b^2\\).

Explain why \\(\\operatorname{Re}(z_1z_2)\\) is not usually \\(\\operatorname{Re}(z_1)\\operatorname{Re}(z_2)\\).