Find the conjugate of \\(3+4i\\).

Find the conjugate of \\(-2-7i\\).

Find the conjugate of \\(5\\).

Find the conjugate of \\(-6i\\).

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

Find \\((4+i)(4+i)^*\\).

Find \\((1-5i)^*-(1-5i)\\).

If \\(z=3e^{i\theta}\\), write \\(z^*\\) in polar form.

Verify \\((z+w)^*=z^*+w^*\\) for \\(z=1+2i\\), \\(w=3-5i\\).

Verify \\((zw)^*=z^*w^*\\) for \\(z=1+i\\), \\(w=2-i\\).

Show that \\(z+z^*=2\\operatorname{Re}(z)\\) for \\(z=-4+9i\\).

Show that \\(z-z^*=2i\\operatorname{Im}(z)\\) for \\(z=-4+9i\\).

Find real \\(t\\) if \\((t+2i)^*=5-2i\\).

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

For which real \\(t\\) is \\((t^2-1+ti)^*=0\\)?

For which real \\(t\\) is \\((t+3i)^*=t+3i\\)?

Find all real \\(t\\) such that \\((t^2-4+ (t-2)i)^*=0\\).

A learner says the conjugate of \\(2-5i\\) is \\(-2+5i\\). Diagnose the error.

Prove that \\((z^*)^*=z\\) for \\(z=a+bi\\).

Explain why multiplying by a conjugate helps divide complex numbers.