Simplifying Radicals Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Simplifying Radicals.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Simplifying a radical means rewriting it so no perfect-square factor remains under the root sign. For example, √50 = √(25·2) = 5√2. The result — called simplified radical form — has the smallest possible number under the radical.

Look inside the radical for perfect squares hiding as factors. $\sqrt{72}$ contains $36 \times 2$ , and since $\sqrt{36} = 6$ , you can pull the 6 out: $\sqrt{72} = 6\sqrt{2}$ . Think of it as freeing numbers that are 'ready' to leave the radical.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Rewrite a radical so the largest perfect-square factor escapes the root and the radicand is as small as possible.

Common stuck point: The procedure for simplifying radicals is the easy part; the trap is using a non-perfect-square factor. Asking "Does the number under the root have any perfect-square factor bigger than 1?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does the number under the root have any perfect-square factor bigger than 1?

Worked Examples

Example 1

easy

Simplify

\sqrt{72}

Answer

6\sqrt{2}

First step

Step 1: Find the largest perfect square factor:

72 = 36 \times 2

Full solution

2
Step 2: $\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \cdot \sqrt{2} = 6\sqrt{2}$ .
3
Check: $6^2 \times 2 = 72$ ✓

To simplify a radical, factor the radicand into a perfect square times a remaining factor. Extract the square root of the perfect square part. The goal is to have no perfect square factors under the radical.

Example 2

medium

Simplify

\sqrt{50x^4y^3}

Example 3

medium

Simplify

\sqrt{200x^4y^3}

Example 4

medium

Simplify

\sqrt{98}

Example 5

medium

Simplify

\sqrt{12x^3y^2}

for

x, y \ge 0

Example 6

medium

Rationalize the denominator:

\dfrac{5}{\sqrt{3}}

Example 7

medium

Multiply:

\sqrt{6} \cdot \sqrt{10}

Example 8

medium

Simplify

\sqrt{75x^2}

for

x \ge 0

Example 9

hard

Rationalize the denominator:

\dfrac{4}{3 - \sqrt{2}}

Example 10

hard

Simplify

\sqrt{50} + \sqrt{18} - \sqrt{8}

Example 11

hard

Rationalize and simplify

\dfrac{6}{\sqrt{12}}

Example 12

challenge

Simplify

\sqrt{9 + 4\sqrt{5}}

into the form

a + b\sqrt{5}

with

a, b

positive integers.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Simplify

\sqrt{48}

Example 2

medium

Simplify

\sqrt{\frac{18}{25}}

Example 3

easy

Simplify

\sqrt{12}

Example 4

easy

Simplify

\sqrt{50}

Example 5

easy

Simplify

\sqrt{18}

Example 6

easy

Simplify

\sqrt{75}

Example 7

easy

Simplify

\sqrt{32}

Example 8

easy

Simplify

\sqrt{x^2}

assuming

x\ge0

Example 9

easy

Simplify

\sqrt{x^6}

assuming

x\ge0

Example 10

easy

Simplify

\sqrt{48}

Example 11

medium

Simplify

\sqrt{72}

Example 12

medium

Simplify

\sqrt{50x^3}

assuming

x\ge0

Example 13

medium

Simplify

3\sqrt{8} + \sqrt{18}

Example 14

medium

Simplify

\sqrt{\frac{49}{16}}

Example 15

medium

Simplify

\sqrt{200}

Example 16

medium

Simplify

\sqrt{45} + \sqrt{20}

Example 17

medium

Simplify

\sqrt{98x^2}

assuming

x\ge0

Example 18

medium

Simplify

\sqrt{300}

Example 19

medium

Simplify

\sqrt{27x^4}

assuming

x\ge0

Example 20

challenge

Simplify

\sqrt[3]{54}

Example 21

challenge

Simplify

\sqrt{72x^5y^2}

assuming

x,y\ge0

Example 22

challenge

Simplify

\sqrt{x^2 - 6x + 9}

assuming

x\ge3

Example 23

easy

Simplify

\sqrt{8}

Example 24

easy

Simplify

\sqrt{45}

Example 25

easy

Simplify

\sqrt{20}

Example 26

easy

Simplify

\sqrt{100x^2}

for

x \ge 0

Example 27

easy

Simplify

\sqrt{x^4}

for

x \ge 0

Example 28

medium

Simplify

\sqrt{x^5}

for

x \ge 0

Example 29

medium

Simplify

3\sqrt{8} + \sqrt{18}

Example 30

medium

Simplify

\sqrt{\dfrac{49}{16}}

Example 31

medium

Simplify

5\sqrt{27} - 2\sqrt{12}

Example 32

medium

Multiply

(2\sqrt{3})(5\sqrt{6})

Example 33

hard

Simplify

(\sqrt{5} + \sqrt{2})^2

Example 34

hard

Simplify

\dfrac{\sqrt{72}}{\sqrt{8}}

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Related Concepts

Square Roots Factors Radical Operations Rationalizing Denominators

Background Knowledge

These ideas may be useful before you work through the harder examples.

square rootsfactors