Once you can simplify square roots, you can add, subtract and multiply them. The rules are ones you know: combining like terms, the distributive property, and the exponent rules.
Why it works
Think of $\sqrt{3}$ as a unit, like $x$. Then $2\sqrt{3} + 5\sqrt{3} = 7\sqrt{3}$ for the same reason $2x + 5x = 7x$. But $\sqrt{2}$ and $\sqrt{3}$ are different units, like $x$ and $y$, so $\sqrt{2} + \sqrt{3}$ cannot be combined. Multiplication has no such limit, because $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$: $\sqrt{2} \cdot \sqrt{3} = \sqrt{6}$.
Why not just use a calculator's decimals? Because $\sqrt{2} = 1.41421356\ldots$ never ends, so any decimal is slightly off, and the errors grow as you keep computing. Radical form is exact. It also shows structure: $\sqrt{50} - \sqrt{18}$ looks hard, but simplified it is $5\sqrt{2} - 3\sqrt{2} = 2\sqrt{2}$, a clean exact answer.
Key idea
Add or subtract only like radicals (same number inside, after simplifying). Multiply with $\sqrt{a}\sqrt{b} = \sqrt{ab}$, multiplying the numbers outside separately, then simplify.
Worked examples
Worked example: Simplify, then combine
Simplify $4\sqrt{3} + \sqrt{12}$.
- $\sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3}$.
- Now they are like terms: $4\sqrt{3} + 2\sqrt{3} = 6\sqrt{3}$.
Answer: $6\sqrt{3}$
Worked example: Multiply
Simplify $2\sqrt{6} \cdot 3\sqrt{10}$.
- Outside numbers: $2 \cdot 3 = 6$. Inside: $\sqrt{6 \cdot 10} = \sqrt{60}$.
- $\sqrt{60} = \sqrt{4 \cdot 15} = 2\sqrt{15}$.
- $6 \cdot 2\sqrt{15} = 12\sqrt{15}$.
Answer: $12\sqrt{15}$
Worked example: Distribute and use special products
Simplify $(3 + \sqrt{2})(3 - \sqrt{2})$ and $(1 + \sqrt{5})^2$.
- The first is a sum times a difference: $3^2 - \left(\sqrt{2}\right)^2 = 9 - 2 = 7$.
- The second is a square of a sum: $1 + 2\sqrt{5} + 5 = 6 + 2\sqrt{5}$.
Answer: $7$ and $6 + 2\sqrt{5}$.
Common mistake
Adding the numbers inside: $\sqrt{2} + \sqrt{3} = \sqrt{5}$. Check with decimals: $\sqrt{2} + \sqrt{3} \approx 1.41 + 1.73 = 3.14$, while $\sqrt{5} \approx 2.24$. Unlike radicals stay as a sum, just like $x + y$.