# Confusion surrounding matrix multiplication rules used in proof

Let $$\mathbf{Z}$$ be the design matrix of a linear regression.

Where $$\mathbf{Z}^{T}\mathbf{Z}$$ is symmetric.

Set $$\mathbf{V} = (\mathbf{Z'Z})^{\frac{1}{2}}(E(\hat{\beta}) - \beta)$$

Where $$\hat{\beta} , \beta$$ is the true and estimated parameter vectors of a linear regression model.

Thus we can say that:

$$Cov(\mathbf{V}) = (\mathbf{Z^{T}\:Z})^{\frac{1}{2}} Cov(\hat{{\bf{\beta}}}) ((\mathbf{Z^{T}\:Z})^{\frac{1}{2}})^{T} =$$

$$(\mathbf{Z^{T}\:Z})^{\frac{1}{2}}\sigma^{2}(\mathbf{Z^{T}\:Z})^{-1}(\mathbf{Z^{T}\:Z})^{\frac{1}{2}} = \sigma^{2}\mathbf{I}$$

Why is this? and from what matrix algebra rule does it follow?

$$\mathbf{A}^{a}\mathbf{A}^{b} = \mathbf{A}^{a + b}$$
$$\mathbf{A}^{0.5}\mathbf{A}^{0.5}\mathbf{A}^{-1} = \mathbf{A}^{0} = \mathbf{I}$$