What are the first three common multiples of these numbers?

Since $360 = 12 \times 30$, 360 is a multiple of 12.

$$\text{LCM}(12, 360) = 360$$

Find the first three multiples of 360.

$$1 \times 360 = 360$$ $$2 \times 360 = 720$$ $$3 \times 360 = 1080$$

The first three common multiples are $\boxed{\text{360, 720, 1080}}$.

2. Find the L.C.M of the Following:

a) 180 and 360

$$180 = 2^2 \times 3^2 \times 5^1$$ $$360 = 2^3 \times 3^2 \times 5^1$$

$$\text{LCM}(180, 360) = 2^{\max(2,3)} \times 3^{\max(2,2)} \times 5^{\max(1,1)}$$ $$\text{LCM}(180, 360) = 2^3 \times 3^2 \times 5^1$$

$$\text{LCM}(180, 360) = 8 \times 9 \times 5 = 72 \times 5 = 360$$

The LCM is $\boxed{360}$.

3. Write in index form the L.C.M of:

a) 40, 60 and 80

$$40 = 2^3 \times 5^1$$ $$60 = 2^2 \times 3^1 \times 5^1$$ $$80 = 2^4 \times 5^1$$

$$\text{LCM}(40, 60, 80) = 2^{\max(3,2,4)} \times 3^{\max(0,1,0)} \times 5^{\max(1,1,1)}$$ $$\text{LCM}(40, 60, 80) = 2^4 \times 3^1 \times 5^1$$

The LCM in index form is $\boxed{2^4 \times 3^1 \times 5^1}$.