\( ( - 8 ) ^ { \beta } = \square _ { i } \)

The given equation is \( (-8)^{\beta} = \square_i \). To solve for \(\beta\), we can use the fact that \((-8)^{\beta}\) can be expressed as \((-8)^{\beta} = (-1)^{\beta} \cdot 8^{\beta}\). Since \((-1)^{\beta}\) can only be equal to 1 or -1, we have two possible cases: 1. \((-1)^{\beta} = 1\) 2. \((-1)^{\beta} = -1\) Let's solve for \(\beta\) in both cases. Solve the equation by following steps: - step0: Solve for \(\beta\): \(\left(-1\right)^{\beta }=1\) - step1: Rewrite in exponential form: \(\left(-1\right)^{\beta }=\left(-1\right)^{0}\) - step2: Set the exponents equal: \(\beta =0\) It seems that there is no real solution for \((-1)^{\beta} = -1\). Therefore, the only solution for \(\beta\) is \(\beta = 0\).