Manipulating Surds

Product Rule: The product of the roots is equal to the root of the product, expressed as $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$, which allows for the decomposition of large roots into manageable factors.

Quotient Rule: Similarly, the division of two square roots can be combined under a single radical, $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$, facilitating the simplification of fractional irrational expressions.

Linear Combination: Surds can only be added or subtracted when they are like terms, meaning the number under the radical (the radicand) is identical, much like combining $3x$ and $5x$ in algebra.

Extracting Square Factors: To simplify $\sqrt{n}$, factorize $n$ into $a^2 \times b$, then apply the product rule to move the square root of the perfect square outside the radical, resulting in $a\sqrt{b}$.

Finding the Largest Square: When multiple square factors exist (e.g., 4 and 16 for 32), always select the highest perfect square to ensure the surd is in its simplest, irreducible form.

Collecting Like Surds: Identify terms with the same irrational part (e.g., $k\sqrt{7}$) and sum their coefficients while keeping the radicand constant, similar to grouping like terms in polynomial addition.

Check for Simplification: Always scan your final answer for potential square factors remaining under the root; an answer like $\sqrt{20}$ will often lose marks if not converted to $2\sqrt{5}$.

Verify via Squaring: To check if $a\sqrt{b}$ is correct, square the outside number and multiply it by the inside number ($a^2 \times b$); the result should equal your original radicand.

Search for Primes: If a radicand is a prime number, such as $\sqrt{13}$ or $\sqrt{17}$, it cannot be simplified further, allowing you to stop looking for factors immediately.

The Addition Fallacy: A very frequent error is assuming $\sqrt{2} + \sqrt{3} = \sqrt{5}$. This is fundamentally incorrect because the radical operation does not distribute over addition or subtraction.

Incomplete Extraction: Students often extract a small square factor but miss a larger one (e.g., simplifying $\sqrt{48}$ to $2\sqrt{12}$ instead of $4\sqrt{3}$), leading to non-standard and often penalized results.

Coefficient Confusion: Ensure that when multiplying terms like $2\sqrt{3} \times 4\sqrt{5}$, you multiply coefficients with coefficients ($8$) and surds with surds ($\sqrt{15}$) separately.