If $M$ is a Riemannian manifold with $\Delta $ its Laplacian, how can we define $(1-\Delta)^{1/2}$?

The book I am reading says that $(1-\Delta)^{1/2}$ is an invertible first-order pseudo-differential operator with the inverse $(1-\Delta)^{-1/2}$. Naively, I try to construct $(1+|\xi|^2)^{1/2}$ as its principal symbol, but I cannot find an operator whose square is $(1-\Delta)$.

Another possible way to define $(1-\Delta)^{1/2}$ is to use the spectral decomposition of $1-\Delta$. If $\{\phi_i\}$ is an orthonormal basis of $L^2(M)$ such that $(1-\Delta)\phi_i=\lambda_i\phi_i$, then we should have $(1-\Delta)^{1/2}\phi_i=\lambda_i^{1/2}\phi_i$. But I cannot see why this is a pseudo-differential operator since I cannot check the definition locally.