Simple Manipulation of Surds

Understanding Surds

Surds refer to root expressions, typically square roots, where the result is an irrational number i.e. it cannot be expressed as a neat fraction.
Common surds include √2, √3, and √5. These are irrational numbers that cannot be precisely represented as rational numbers, but can be approximated to decimal places.
The Rationalising the Denominator rule is used to remove surds from the denominator of a fraction. For example, to rationalise 1/√3, multiply both the numerator and the denominator by √3 to get √3/3.

Rules of Surds

Applying the product rule, the square root of a product equals the product of the square roots: √(ab) = √a * √b.
Applying the quotient rule, the square root of a quotient equals the quotient of the square roots: √(a/b) = √a / √b.
When simplifying surds, apply the rule of simplifying which states that √(a²b) = a√b for a > 0.

Operations involving Surds

Surds can be added and subtracted only if they involve the same radicand (the expression under the root). For example, √3 + 2√3 = 3√3.
Surds can be multiplied and divided according to the standard rules of multiplication and division by applying the product and quotient rules of surds.

Simplification of Surds

Surds can often be simplified by finding the highest square factor of the number under the root. Divide the number by this square and write it as a product under the root.
The process of simplification seeks to express the surd in the most simplified or smallest root form. For instance, √50 can be simplified to 5√2.

Remember, understanding surds and being able to manipulate them is a useful skill not only for handling logarithm and indices, but also for higher levels of maths. Regular practice is essential in achieving proficiency.