Elasticity of substitution production function

Question

Consider the given general form of a constant elasticity of substitution production function:

y = [δL^ρ + (1-δ)k^ρ]^1/ρ   

You may find this question easier to answer if you let α₁ = δ^1/ρ and α₂ = (1−δ)^1/ρ.  By all means, leave your answer in terms of α1 and α2.  Assume a firm is trying to minimize the cost of producing any given y.  Costs are given by

C = wL + rK.

Find the firm’s cost minimizing demand function for L. The cost minimizing demand for K is determined simultaneously (so you need both FOCs) but since I am sure many of you left this until the last minute, you only have to come up with the expression for L. However, for your own practice, you may want to find the equivalent expression for K on your own.  You may assume that nonnegativity constraints on L and K are not binding.