Simplify the following expressions involving square roots, showing two methods for each:

Here are the solutions for each part:

a) $\sqrt{64 \times 36}$

Method 1: Calculate the product first, then take the square root.

$$\sqrt{64 \times 36} = \sqrt{2304} = 48$$

Method 2: Take the square root of each number, then multiply.

$$\sqrt{64} \times \sqrt{36} = 8 \times 6 = 48$$

The statements give the same answer. This is because $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$.

b) $\sqrt{169 - 25}$

Method 1: Calculate the difference first, then take the square root.

$$\sqrt{169 - 25} = \sqrt{144} = 12$$

Method 2: Take the square root of each number, then subtract.

$$\sqrt{169} - \sqrt{25} = 13 - 5 = 8$$

The statements give different answers. This is because $\sqrt{a - b} \neq \sqrt{a} - \sqrt{b}$.

c) $\sqrt{64 \div 16}$

Method 1: Calculate the division first, then take the square root.

$$\sqrt{64 \div 16} = \sqrt{4} = 2$$

Method 2: Take the square root of each number, then divide.

$$\sqrt{64} \div \sqrt{16} = 8 \div 4 = 2$$

The statements give the same answer. This is because $\sqrt{a \div b} = \sqrt{a} \div \sqrt{b}$.