$$\begin{gathered}2\sqrt{5} + 7\sqrt{5} = 9\sqrt{5}\\[3pt] \sqrt{a} \times \sqrt{b} = \sqrt{ab}\\[3pt] 2\sqrt{3} \times 5\sqrt{7} = 10\sqrt{21}\\[3pt] \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\end{gathered}$$

Add/subtract only like surds; for ×/÷ pair numbers with numbers, surds with surds. Reflex: A surd times itself equals the number inside — $\sqrt2\times\sqrt2=2$.

$$\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2}$$

Spot a perfect square factor inside the surd, pull it out. This turns hidden matches into 'like' surds you can add. Reflex: Press calculator to check $\sqrt{18}$ and $3\sqrt{2}$ give the same value.

$$\begin{gathered}\frac{2}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{3}}{3}\\[3pt] \frac{10}{3\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{10\sqrt{5}}{15} = \frac{2\sqrt{5}}{3}\end{gathered}$$

One surd on the bottom? Multiply top and bottom by that surd. A number in front ($3\sqrt{5}$) stays put — only the surd needs clearing. Reflex: You're only multiplying by $1$ — same value, new form.

$$\begin{gathered}(a+b)(a-b) = a^2 - b^2\\[3pt] (\sqrt{5}+\sqrt{3})(\sqrt{5}-\sqrt{3}) = 5 - 3 = 2\end{gathered}$$

Two-term denominator: multiply top and bottom by the conjugate (same pair, middle sign flipped) — difference-of-squares wipes out the surd below.

Reflex: Conjugate = just flip the middle sign.

Surd in denominator → rationalize. Big surd anywhere → break into smaller surds. Do both, then collect like terms.

Reflex: Simplify the surds first — small surds make the like terms visible.

$$\begin{gathered}p + q\sqrt{n} = r + s\sqrt{n}\\[3pt] \Rightarrow\ p = r \ \text{ and } \ q = s\end{gathered}$$

Group each side into (number) + (number)$\sqrt{n}$ shape first. A rational can never equal an irrational — that's why they split. Reflex: Match rational parts, then match surd parts, separately.

$$\begin{gathered}x\sqrt{3} + 5 = 8x\\[2pt] x\sqrt{3} - 8x = -5\\[2pt] x(\sqrt{3} - 8) = -5\\[2pt] x = \frac{-5}{\sqrt{3} - 8}\end{gathered}$$

The surds are just coefficients: group every $x$-term on one side, factorize $x$ out, divide — then rationalize the fraction.

Reflex: Group → factorize → divide → rationalize.

$$\begin{gathered}\sqrt{x+5} - 2 = \sqrt{3}\\[2pt] \sqrt{x+5} = \sqrt{3} + 2\\[2pt] x + 5 = (\sqrt{3} + 2)^2\end{gathered}$$

To remove a square root, square both sides — but isolate the root first, so one squaring clears it. Reflex: Check answers in the original — squaring can create false roots.