Rules of Surds

Laws of Surds

• Basic Multiplication Rule:
  \sqrt{a} \times \sqrt{b} = \sqrt{ab}

• Squaring a Surd:
  \sqrt{a} \times \sqrt{a} = \sqrt{a^2} = a

• Multiplying Coefficients and Surds:
  a\sqrt{b} \times c\sqrt{d} = ac\sqrt{bd}

• Dividing by a Surd:
  \frac{a}{\sqrt{b}} = \frac{a\sqrt{b}}{b}

• Multiplying Identical Surds:
  \sqrt{b} \times \sqrt{b} = b

Rules for Multiplying Surds

• Coefficient times Surd:
  a \times \sqrt{b} = a\sqrt{b}

  • Example:
    3 \times \sqrt{2} = 3\sqrt{2}

• Surd times Identical Surd:
  \sqrt{b} \times \sqrt{b} = b

  • Example:
    \sqrt{3} \times \sqrt{3} = 3

• Surd times Surd:
  \sqrt{a} \times \sqrt{b} = \sqrt{ab}

  • Example:
    \sqrt{2} \times \sqrt{3} = \sqrt{6}

• Multiplying Coefficients and Surds:
  a\sqrt{b} \times c\sqrt{d} = ac\sqrt{bd}

  • Example:
    2\sqrt{3} \times 2\sqrt{5} = 4\sqrt{15}