Manipulation of Surds, including Rationalising the Denominator

Manipulation of Surds, including Rationalising the Denominator

Understanding Surds

• Surds are numbers left in ‘square root form’ (or ‘cube root form’ etc). They are therefore irrational numbers.
• Surds are used when exact values are needed.

Manipulation of Surds

• When manipulating surds, remember that you can only add or subtract like surds, just as you can only add or subtract like terms.
• Multiplying surds: Suppose √a and √b are two surds. The multiplication of these two surds is given by √(a*b).

Simplifying Surds

• To simplify a surd, write the number under the square root sign as the product of two factors, one of which is the largest perfect square.

Rationalising the denominator

• Rationalising the denominator is a process used to eliminate the surds from the denominator of a fraction.
• For a single term denominator, multiply both numerator and denominator by the surd. For a two term denominator, use difference of squares.
• For example to rationalise 1/√2, multiply both numerator and denominator by √2 to get √2/2.

Operations involving Surds

• Addition, subtraction, multiplication and division can be performed on surds along with rationalisation.
• To add or subtract surds, they must be like surds, means that the number under the root must be the same.

These notes should guide you through the process of understanding, manipulating, and rationalising surds. Always practise applying these principles in various problems to cement your understanding.