The numbers that can be expressed in fractional form \(\dfrac{a}{b}\) where \(a\) and \(b\) are integers and \(b \ne 0\).

The numbers that cannot be expressed in fractional form.

\(\sqrt{a}\times \sqrt{b}=\sqrt{a\times b}\)

\(\dfrac{\sqrt{a}}{\sqrt{b}}=\sqrt{\dfrac{a}{b}}\)

\(k\times \sqrt{a}=\sqrt{k^2\times a}\) (for \(k\) is a rational number)

\((\sqrt{a})^2=a\)

• \(\sqrt{ab}= \sqrt{a} \times \sqrt{b}\)

• \(\sqrt{a} \times \sqrt{b}=\sqrt{ab}\)

• To eliminate surds from the denominator of a fraction.
• Example:
  \(\dfrac{1}{\sqrt{a}}\times \dfrac{\sqrt{a}}{\sqrt{a}}=\dfrac{\sqrt{a}}{a}\)

• Use the conjugate.
• Example:
  \(\dfrac{1}{a+\sqrt{b}} \times \dfrac{a-\sqrt{b}}{a-\sqrt{b}}=\dfrac{a-\sqrt{b}}{a^2-b}\)

• Isolate the Surd: Separate the surd on one side of the equation. 
• Square Both Sides: Square both sides of the equation to eliminate the surd.
• Solve the Equation: Solve the resulting linear or quadratic equation.

• Only identical surds (same root and radicand) can be added or subtracted.
• Example:
  \(3\sqrt{2}+2\sqrt{2}=5\sqrt{2}\)

• Use the laws of multiplication and division of surds.

Solve \(x-4\sqrt{x}+3=0\).

Use the factorisation method.

\(\begin{aligned} x-4\sqrt{x}+3&=0 \\ (\sqrt{x}-3)(\sqrt{x}-1)&=0 \end{aligned}\)

\(\begin{aligned} \sqrt{x}-3&=0 \\ \sqrt{x}&=3 \\ (\sqrt{x})^2&=3^2 \\ x&=9 \end{aligned}\) or \(\begin{aligned} \sqrt{x}-1&=0 \\ \sqrt{x}&=1 \\ (\sqrt{x})^2&=1^2 \\ x&=1.\end{aligned}\)

Thus, \(x=9\) and \(x=1\) are the solutions for the equation.

The numbers that can be expressed in fractional form \(\dfrac{a}{b}\) where \(a\) and \(b\) are integers and \(b \ne 0\).

The numbers that cannot be expressed in fractional form.

\(\sqrt{a}\times \sqrt{b}=\sqrt{a\times b}\)

\(\dfrac{\sqrt{a}}{\sqrt{b}}=\sqrt{\dfrac{a}{b}}\)

\(k\times \sqrt{a}=\sqrt{k^2\times a}\) (for \(k\) is a rational number)

\((\sqrt{a})^2=a\)

• \(\sqrt{ab}= \sqrt{a} \times \sqrt{b}\)

• \(\sqrt{a} \times \sqrt{b}=\sqrt{ab}\)

• To eliminate surds from the denominator of a fraction.
• Example:
  \(\dfrac{1}{\sqrt{a}}\times \dfrac{\sqrt{a}}{\sqrt{a}}=\dfrac{\sqrt{a}}{a}\)

• Use the conjugate.
• Example:
  \(\dfrac{1}{a+\sqrt{b}} \times \dfrac{a-\sqrt{b}}{a-\sqrt{b}}=\dfrac{a-\sqrt{b}}{a^2-b}\)

• Isolate the Surd: Separate the surd on one side of the equation. 
• Square Both Sides: Square both sides of the equation to eliminate the surd.
• Solve the Equation: Solve the resulting linear or quadratic equation.

• Only identical surds (same root and radicand) can be added or subtracted.
• Example:
  \(3\sqrt{2}+2\sqrt{2}=5\sqrt{2}\)

• Use the laws of multiplication and division of surds.

Solve \(x-4\sqrt{x}+3=0\).

Use the factorisation method.

\(\begin{aligned} x-4\sqrt{x}+3&=0 \\ (\sqrt{x}-3)(\sqrt{x}-1)&=0 \end{aligned}\)

\(\begin{aligned} \sqrt{x}-3&=0 \\ \sqrt{x}&=3 \\ (\sqrt{x})^2&=3^2 \\ x&=9 \end{aligned}\) or \(\begin{aligned} \sqrt{x}-1&=0 \\ \sqrt{x}&=1 \\ (\sqrt{x})^2&=1^2 \\ x&=1.\end{aligned}\)

Thus, \(x=9\) and \(x=1\) are the solutions for the equation.