Multiplying and Dividing Square Roots

• Multiplication Rule
  If $ a \geq 0 $ and $ b \geq 0 $, then:
  $$ \sqrt{a} \times \sqrt{b} = \sqrt{a \times b} $$
• Division Rule
  If $ a \geq 0 $ and $ b > 0 $, then:

$$ \sqrt{a} \div \sqrt{b} = \sqrt{a \div b} $$

or equivalently,

$$ \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} $$

By following these rules, you can efficiently work with square roots in calculations.

The multiplication rule states that for any non-negative numbers $ a $ and $ b $:

$$ \sqrt{a} \times \sqrt{b} = \sqrt{a \times b} $$

By applying this rule, we can simplify square roots effectively.

• Example 1:
  $$ \sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2} $$
• Example 2:
  First, we apply the multiplication rule:

$$ \sqrt{4.8} \times \sqrt{10} = \sqrt{4.8 \times 10} = \sqrt{48} $$

Now, we reverse the rule to simplify:

$$ \sqrt{48} = \sqrt{16 \times 3} = \sqrt{16} \times \sqrt{3} = 4\sqrt{3} $$

This example is useful because it demonstrates the square root multiplication rule in both directions.

The division rule states that for any non-negative numbers $ a $ and positive number $ b $:

$$ \sqrt{a} \div \sqrt{b} = \sqrt{a \div b} $$

or equivalently,

$$ \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} $$

By applying this rule, we can simplify square roots effectively.

• Example 1:
  $$ \sqrt{36} \div \sqrt{25} = \sqrt{36 \div 25} = 6 \div 5 = 1.2 $$
• Example 2:
  $$ \frac{\sqrt{50}}{\sqrt{2}} = \sqrt{\frac{50}{2}} = \sqrt{25} = 5 $$