Surds — Simplifying, Adding & Rationalizing

Surds are root expressions that are still irrational after simplification, such as $\sqrt{2}$ and $3\sqrt{5}$. The standard workflow is the same every time: simplify first, combine only like surds, and rationalize the denominator when a root remains there.

When Something Counts As A Surd

Use surd techniques when a root does not reduce to a rational number. If a root simplifies to a rational, it is not usually treated as a surd: for example,

so $\sqrt{9}$ is not a surd once simplified. But $\sqrt{2}$ does not simplify to a rational number, so it is a surd. The same goes for $2\sqrt{3}$, $\sqrt{7}$, or $5\sqrt{11}$: exact radical expressions whose simplified values stay irrational. Surds matter because they keep exact values, and $\sqrt{2}$ is more precise than a rounded decimal such as $1.414$.

They appear in geometry, the Pythagorean theorem, quadratic equations, trigonometry, and algebraic simplification, wherever exact form matters more than a decimal. A square with side length $3$ has diagonal $3\sqrt{2}$, not just an approximation.

The Steps

1. Simplify the root first. Factor the number inside the root and pull out any perfect-square factors before doing anything else.
2. Rewrite in simplest surd form. Turn expressions like $\sqrt{72}$ into $6\sqrt{2}$ so the radical part is easier to compare.
3. Combine only like surds. Add or subtract surds only when their simplified radical parts match, such as $4\sqrt{2} + 3\sqrt{2}$.
4. Rationalize the denominator if needed. If a root remains in the denominator, multiply by a suitable factor so the denominator becomes rational.
5. Keep the expression exact. Leave the result in surd form unless the problem asks for a decimal approximation.

To simplify, look for a perfect-square factor inside the root, for example

If the number under the root has no perfect-square factor greater than $1$, the surd is already simplified. You can add or subtract only like surds; for instance, $2\sqrt{8} + \sqrt{18}$ first simplifies to $4\sqrt{2}$ and $3\sqrt{2}$, which combine to $7\sqrt{2}$.

To rationalize a single-surd denominator, multiply top and bottom by that surd:

If the denominator has two terms such as $a + \sqrt{b}$, use the conjugate $a - \sqrt{b}$, because $(a + \sqrt{b})(a - \sqrt{b}) = a^2 - b$ has no surd term.

Full Worked Example: Simplify, Add, Then Rationalize

Simplify

Simplify each root.

Combine like surds in the numerator.

Rationalize the denominator by multiplying top and bottom by $\sqrt{3}$:

Keep it exact. The simplified result is

Practice With A Built-In Self-Check

Try simplifying

Work the steps in order. Self-check: $\sqrt{50} = 5\sqrt{2}$ and $\sqrt{8} = 2\sqrt{2}$, so the numerator becomes $7\sqrt{2}$, the fraction becomes $\frac{7\sqrt{2}}{\sqrt{2}} = 7$, and no rationalization is needed because the surds cancel exactly. If you reach a messier answer, recheck whether you simplified each root before dividing.

Before trusting any surd answer, run four quick checks: have I simplified every surd first; are the surds actually like terms before adding; does any denominator still contain a root; and am I keeping the answer exact unless asked for a decimal?

Where Each Step Goes Wrong

Adding before simplifying. $\sqrt{8}$ and $\sqrt{18}$ are not obviously like surds, but they become $2\sqrt{2}$ and $3\sqrt{2}$ after simplification. Skip that step and you miss an easy combination.

Combining unlike surds. In general $\sqrt{2} + \sqrt{3} \ne \sqrt{5}$. You can only combine coefficients when the simplified radical parts match.

Splitting roots across addition. In general $\sqrt{a + b} \ne \sqrt{a} + \sqrt{b}$. For example $\sqrt{4 + 5} = \sqrt{9} = 3$, but $\sqrt{4} + \sqrt{5} = 2 + \sqrt{5}$, which is not $3$.

Cancelling radicals incorrectly. In $\frac{\sqrt{2}}{\sqrt{3}}$ you cannot cancel the square roots because the numbers inside differ; simplify correctly or rationalize instead.