Surds — KCSE Mathematics

Surds are irrational numbers that cannot be expressed as exact fractions. To simplify expressions involving surds, follow these steps:

1. Identify the surd: Look for square roots or cube roots in the expression.
2. Factor out perfect squares: For square roots, find the largest perfect square that divides the number under the root.
3. Simplify: Rewrite the surd using the factors identified.

For example, to simplify ( \sqrt{50} ):

• Factor 50: ( 50 = 25 \times 2 )
• Identify the perfect square: ( 25 ) is a perfect square.
• Rewrite the expression: ( \sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2} )

Another example is simplifying ( \sqrt{72} ):

• Factor 72: ( 72 = 36 \times 2 )
• Identify the perfect square: ( 36 ) is a perfect square.
• Rewrite the expression: ( \sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2} )

Practice simplifying various surds to gain confidence!

To rationalise a denominator containing a surd, we eliminate the surd from the denominator. This is done by multiplying both the numerator and the denominator by the conjugate of the denominator.

Steps to Rationalise:

1. Identify the surd in the denominator.
2. Multiply the numerator and denominator by the conjugate of the denominator.
3. Simplify the expression.

Example: Rationalise ( \frac{3}{\sqrt{2}} ).

• Multiply by ( \frac{\sqrt{2}}{\sqrt{2}} ):
  ( \frac{3 \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} = \frac{3\sqrt{2}}{2} ).

Another Example: Rationalise ( \frac{5}{2 + \sqrt{3}} ).

• Multiply by ( \frac{2 - \sqrt{3}}{2 - \sqrt{3}} ):
  ( \frac{5(2 - \sqrt{3})}{(2 + \sqrt{3})(2 - \sqrt{3})} = \frac{10 - 5\sqrt{3}}{4 - 3} = 10 - 5\sqrt{3} ).

Rationalising helps in simplifying calculations and makes the expression easier to handle.