Surds — Simplifying, Adding & Rationalizing

Surds are root expressions that are still irrational after simplification. Typical examples are $\sqrt{2}$ and $3\sqrt{5}$. To handle surds, simplify first, combine only like surds, and rationalize the denominator when a root remains there.

Surds matter because they keep exact values. For example, $\sqrt{2}$ is more precise than a rounded decimal such as $1.414$.

What A Surd Means

If a root simplifies to a rational number, it is not usually treated as a surd. For example,

$$\sqrt{9} = 3$$

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 idea applies to expressions like $2\sqrt{3}$, $\sqrt{7}$, or $5\sqrt{11}$. They are exact radical expressions whose simplified values are still irrational.

How To Simplify Surds

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

For example,

$$\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36}\sqrt{2} = 6\sqrt{2}$$

The goal is to pull perfect squares out of the radical and leave only the non-square part inside.

If the number under the root has no perfect-square factor greater than $1$, the surd is already simplified.

How To Add And Subtract Surds

You can add or subtract surds only when they are like surds, meaning their simplified radical parts are the same.

For example,

$$2\sqrt{8} + \sqrt{18}$$

cannot be combined immediately. First simplify each surd:

$$2\sqrt{8} = 2 \cdot 2\sqrt{2} = 4\sqrt{2}$$

and

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

Now both terms have the same radical part, so

$$4\sqrt{2} + 3\sqrt{2} = 7\sqrt{2}$$

This is the key pattern: simplify first, then combine coefficients if the radical part matches.

Worked Example: Simplify, Add, Then Rationalize

Simplify

$$\frac{2\sqrt{8} + \sqrt{18}}{\sqrt{3}}$$

Start by simplifying the numerator:

$$2\sqrt{8} = 4\sqrt{2}$$

and

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

So the fraction becomes

$$\frac{4\sqrt{2} + 3\sqrt{2}}{\sqrt{3}} = \frac{7\sqrt{2}}{\sqrt{3}}$$

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

$$\frac{7\sqrt{2}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{7\sqrt{6}}{3}$$

So the simplified result is

$$\frac{7\sqrt{6}}{3}$$

This one example shows the full workflow: simplify each surd, combine like surds, then rationalize the denominator.

How To Rationalize The Denominator

Rationalizing a denominator means removing roots from the bottom of a fraction without changing its value.

If the denominator is a single surd, multiply top and bottom by that surd. For example,

$$\frac{5}{\sqrt{3}} = \frac{5}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{5\sqrt{3}}{3}$$

If the denominator has two terms, such as $a + \sqrt{b}$, use the conjugate $a - \sqrt{b}$. That works because

$$(a + \sqrt{b})(a - \sqrt{b}) = a^2 - b$$

and $a^2 - b$ has no surd term.

Common Mistakes With Surds

Adding Before Simplifying

$\sqrt{8}$ and $\sqrt{18}$ are not obviously like surds, but after simplification they become $2\sqrt{2}$ and $3\sqrt{2}$. If you skip the simplification step, you often miss an easy combination.

Combining Unlike Surds

In general,

$$\sqrt{2} + \sqrt{3} \ne \sqrt{5}$$

You can only combine the 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 equal to $3$.

Cancelling Radicals Incorrectly

In

$$\frac{\sqrt{2}}{\sqrt{3}}$$

you cannot cancel the square roots because the numbers inside them are different. You need to simplify correctly or rationalize the denominator.

When Surds Are Used

Surds show up whenever exact values involve roots that are not perfect squares. Common places include geometry, the Pythagorean theorem, quadratic equations, trigonometry, and algebraic simplification.

They are especially useful when exact form matters more than a decimal approximation. For example, a square with side length $3$ has diagonal $3\sqrt{2}$, not just an approximate decimal.

Quick Checklist For Surd Problems

When you work with surds, ask:

1. Have I simplified every surd first?
2. Are the surds actually like terms before I add or subtract them?
3. If there is a fraction, does the denominator still contain a root?
4. Am I keeping the answer exact unless the problem asks for a decimal?

Those four checks prevent most routine errors.

Try A Similar Problem

Try simplifying

$$\frac{\sqrt{50} + \sqrt{8}}{\sqrt{2}}$$

Work in the same order: simplify each surd, combine like terms, then check whether any rationalization is still needed. If you use a step-by-step solver, compare each algebra step rather than only the final answer.