1. The problem is to find the value of $w$ in the equation $$\sqrt{82^2 + 15^2 + 144^2} = \sqrt{82^2 + 15^2 + w^2}.$$\n\n2. First, calculate the squares of the known numbers:\n$$82^2 = 6724,$$\n$$15^2 = 225,$$\n$$144^2 = 20736.$$\n\n3. Sum these values on the left side:\n$$6724 + 225 + 20736 = 27685.$$\n\n4. The equation becomes:\n$$\sqrt{27685} = \sqrt{6724 + 225 + w^2}.$$\n\n5. Simplify the right side sum without $w^2$:\n$$6724 + 225 = 6949,$$\nso the equation is:\n$$\sqrt{27685} = \sqrt{6949 + w^2}.$$\n\n6. Square both sides to eliminate the square roots:\n$$27685 = 6949 + w^2.$$\n\n7. Solve for $w^2$:\n$$w^2 = 27685 - 6949 = 20736.$$\n\n8. Take the square root of both sides to find $w$:\n$$w = \sqrt{20736} = 144.$$\n\n9. Therefore, the value of $w$ is 144.