A swing dies down as x = A e⁻ᵇᵗcos(ω t), where t is time…

Step 1. (a). $bt$ is a pure number, so $[b] = [\text{T}^{-1}]$.

Step 2. (b), the first term. $[a] = \dfrac{[\text{M}\,\text{L}\,\text{T}^{-2}]}{[\text{L}^{1/2}]} = [\text{M}\,\text{L}^{1/2}\,\text{T}^{-2}]$.

Step 3. (b), the second term. $[b] = \dfrac{[\text{M}\,\text{L}\,\text{T}^{-2}]}{[\text{L}^{2}]} = [\text{M}\,\text{L}^{-1}\,\text{T}^{-2}]$.