Q.Simplify the following and express in the form $a+ib$ : $3+\sqrt{-64}$

Complex numbers combine under addition, subtraction, multiplication, division and conjugation using rules that extend ordinary real-number algebra, treating $i$ as a symbol satisfying $i^2=-1$. Addition and subtraction act componentwise: $(a+ib)\pm(c+id)=(a\pm c)+i(b\pm d)$ — real parts combine with real parts, imaginary with imaginary. Scalar multiplication by a real $k$ scales both parts: $k(a+ib)=ka+i(kb)$. Multiplication expands like two binomials and then uses $i^2=-1$ to collapse the $i^2$ term: $(a+ib)(c+id)=ac+adi+bci+bdi^2=(ac-bd)+i(ad+bc)$. All four operations retain the familiar commutative, associative and identity properties from real-number arithmetic. The conjugate of $z=a+ib$ is $\bar z=a-ib$ — obtained by flipping only the sign of the imaginary part — and satisfies $\overline{\overline z}=z$, $z=\bar z$ exactly when $z$ is real, $z=-\bar z$ exactly when $z$ is purely imaginary, and the useful product identity $z\bar z=a^2+b^2$ (always a non-negative real number). The conjugate also distributes cleanly over every operation: $\overline{z_1\pm z_2}=\bar z_1\pm\bar z_2$ and $\overline{z_1 z_2}=\bar z_1\bar z_2$, and similarly for quotients. These rules are the toolkit used throughout the chapter to simplify expressions, verify identities, and reduce any combination of complex numbers to the standard $a+ib$ form.