Calculate the square root of the following numbers: a) 221a36 b) 221a144 c) 221a324 d) 221a1296

$$36 = 2 \times 18$$ $$18 = 2 \times 9$$ $$9 = 3 \times 3$$

So, the prime factorization of 36 is $2 \times 2 \times 3 \times 3$.

$$\sqrt{36} = \sqrt{2 \times 2 \times 3 \times 3}$$ $$\sqrt{36} = \sqrt{2^2 \times 3^2}$$ $$\sqrt{36} = 2 \times 3$$

$$\sqrt{36} = 6$$

The square root of 36 is $\boxed{\text{6}}$.

$$144 = 2 \times 72$$ $$72 = 2 \times 36$$ $$36 = 2 \times 18$$ $$18 = 2 \times 9$$ $$9 = 3 \times 3$$

So, the prime factorization of 144 is $2 \times 2 \times 2 \times 2 \times 3 \times 3$.

$$\sqrt{144} = \sqrt{2 \times 2 \times 2 \times 2 \times 3 \times 3}$$ $$\sqrt{144} = \sqrt{2^2 \times 2^2 \times 3^2}$$ $$\sqrt{144} = 2 \times 2 \times 3$$

$$\sqrt{144} = 12$$

The square root of 144 is $\boxed{\text{12}}$.

$$324 = 2 \times 162$$ $$162 = 2 \times 81$$ $$81 = 3 \times 27$$ $$27 = 3 \times 9$$ $$9 = 3 \times 3$$

So, the prime factorization of 324 is $2 \times 2 \times 3 \times 3 \times 3 \times 3$.

$$\sqrt{324} = \sqrt{2 \times 2 \times 3 \times 3 \times 3 \times 3}$$ $$\sqrt{324} = \sqrt{2^2 \times 3^2 \times 3^2}$$ $$\sqrt{324} = 2 \times 3 \times 3$$

$$\sqrt{324} = 18$$

The square root of 324 is $\boxed{\text{18}}$.

$$1296 = 2 \times 648$$ $$648 = 2 \times 324$$ $$324 = 2 \times 162$$ $$162 = 2 \times 81$$ $$81 = 3 \times 27$$ $$27 = 3 \times 9$$ $$9 = 3 \times 3$$

So, the prime factorization of 1296 is $2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 3$.

$$\sqrt{1296} = \sqrt{2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 3}$$ $$\sqrt{1296} = \sqrt{2^2 \times 2^2 \times 3^2 \times 3^2}$$ $$\sqrt{1296} = 2 \times 2 \times 3 \times 3$$

$$\sqrt{1296} = 36$$

The square root of 1296 is $\boxed{\text{36}}$.

Problem solving: Use any three examples to show that the square of any odd number is always an odd number.

Example 1:
Choose the odd number 1.

$$1^2 = 1 \times 1 = 1$$

Since 1 is not divisible by 2, it is an odd number.

Example 2:
Choose the odd number 3.

$$3^2 = 3 \times 3 = 9$$

Since 9 is not divisible by 2, it is an odd number.

Example 3:
Choose the odd number 5.

$$5^2 = 5 \times 5 = 25$$

Since 25 is not divisible by 2, it is an odd number.

These examples demonstrate that the square of an odd number is always an odd number.