Manipulating Surds

Understand the definition: Surds are square roots which cannot be simplified to a rational number. They are ‘irrational’ numbers and are often left in surd form to maintain precision.
Know how to simplify surds: Simplifying a surd involves removing any-square rootable factors. For instance, √18 can be simplified to 3√2. This is achieved by identifying the highest square number that can divide 18 (which is 9), taking the square root of that number (which is 3), and multiplying it with the square root of the remainder (which is √2).
Remember surd rules: √a x √b equals √(a x b) (multiplication rule), and √(a/b) equals √a / √b (division rule).
Be aware of surd addition and subtraction: You can only add or subtract surds if they are ‘like’ or ‘similar’ i.e., if they have the same number under the square root sign (radicand). For instance, 4√3 + 2√3 = 6√3.
Recognize standard surd forms: The standard form of a surd is to have no square roots in the denominator of fractions. To achieve this, you multiply the numerator and the denominator by the conjugate of the denominator. This is known as ‘rationalizing the denominator’.
Practice expanding brackets with surds: Use the FOIL method (First, Outside, Inside, Last) to expand brackets with surds. Be cautious of signs, and remember that the product of two identical surds is a rational number (as √a x √a equals a).
Brush up on solving equations involving surds: To solve such equations, you often need to square both sides at some point. Be careful with negatives - if (√a = -b) is squared, the result is (a = b^2), not (- a = b^2).
Tackle more complex surds problems: Once confident with basic manipulation and simplifying surds, try tackling problems involving surd expressions in both the numerator and denominator, or equations where the surd quantity is not isolated.
Practice makes perfect: There are a variety of online platforms and textbooks available offering an array of surd problems that will build familiarity and fluency. It’s important to check your answers and understand any mistakes you may have made along the way.

Remember, although surds can seem complex initially, they follow clear rules and with regular practice, you’ll soon find them straightforward to manipulate.