Physics

The key insight is that Pauli matrices satisfy $\sigma_i^2 = I$ and anticommute when different, which means $(\mathbf{c} \cdot \boldsymbol{\sigma})^2 = |\mathbf{c}|^2 I$—the product collapses to a scalar times the identity. This alternating pattern (odd powers give $\mathbf{c} \cdot \boldsymbol{\sigma}$, even powers give $|\mathbf{c}|^2 I$) is your ticket to evaluating any power. For part (ii), the hint suggests expanding with the binomial theorem, then recognizing that the mixed terms reorganize into the same alternating structure, letting you collapse the sum back into a closed form using the binomial theorem in reverse.

The key insight is that Pauli matrices satisfy $\sigma_i^2 = I$ and anticommute when different, which means $(\mathbf{c} \cdot \boldsymbol{\sigma})^2 = |\mathbf{c}|^2 I$—the product collapses to a scalar times the identity. This alternating pattern (odd powers give $\mathbf{c} \cdot \boldsymbol{\sigma}$, even powers give $|\mathbf{c}|^2 I$) is your ticket to evaluating any power. For part (ii), the hint suggests expanding with the binomial theorem, then recognizing that the mixed terms reorganize into the same alternating structure, letting you collapse the sum back into a closed form using the binomial theorem in reverse.