Manipulating surds

Introductory Concepts

Surds are numbers left in ‘square root form’ (or ‘cube root form’ etc). They are often used in calculations to give exact values.
To comprehend surds, it’s essential to understand square roots. The square root of a number is a value that, when squared, gives the original number.
For example, √16 = 4 because 4² = 16. Therefore, four is a root of 16.
When a square root cannot be simplified to a whole number, it is termed as a surd. Example: √3

One may simplify a surd by factoring down its radicand (the number under the root symbol) to a product of numbers where at least one of them have a square root that is a rational number.
For example, to simplify √18, factor it to √9×2. As √9 is a rational number (3), √18 simplifies to 3√2.
When simplified properly, the only number under the square root should not have any factors which are perfect squares except 1.

In mathematics, a fraction is more ‘acceptable’ if its denominator is a rational number. To rationalise the denominator means to remove surds from it.
To rationalise a denominator with a single term, multiply both the top and bottom by the surd. For instance, to rationalize 1/√2, multiply by √2/√2. The result is √2/2.
To rationalise the denominator of a fraction where the denominator is a binomial (2 terms with a surd), multiply the top and bottom by the conjugate. The conjugate is identical to the denominator but with the sign in the middle flipped.
For example, rationalize 2/(3+√2) by multiplying by (3-√2)/(3-√2). The result after simplification is (6-2√2)/7.

Adding or subtracting surds: Surds can only be added or subtracted if they are ‘like’ terms. If two surds have the same number under the root, they can be added or subtracted in the same way algebraic terms can be. For example, √2 + √2 = 2√2, √3 +√2 ≠ 2√5.
Multiplying and dividing surds: To multiply surds together, multiply the numbers under the root symbols and write them under one root symbol. For instance, √3 x √2 = √6. Similarly, to divide, divide the numbers under the root symbols. Example: √8/√2 = √4 = 2.

Precise manipulation of surds often simplifies arithmetic and algebraic operations.
Always keep the rules for surd operations on the tip of your fingers to smoothly navigate more complex surd manipulations tasks.
When in doubt, rationalise the denominator or simplify the surd to smaller ‘components’. This can often reveal hidden patterns or easier routes to the solution.
Regular practice is key to getting proficient at manipulating surds.