How to Simplify Radical Expressions? (+FREE Worksheet!)
Simplifying radical expressions means rewriting them so no perfect-square factors remain under the radical sign. It is a key skill for working with square roots throughout Algebra 1 and is needed for adding, subtracting, multiplying, and solving equations with radicals. This guide covers the technique step by step, with four worked examples, two video lessons, and practice problems.
What Is a Radical Expression?
A radical expression contains a square root (or other root) symbol. The expression under the radical sign is called the radicand. A radical is in simplest form when:
- The radicand has no perfect-square factors other than 1.
- There are no fractions under the radical.
- There are no radicals in the denominator.
How to Simplify Radical Expressions
Step 1: Find the Largest Perfect Square Factor
Factor the radicand so that one factor is the largest perfect square (4, 9, 16, 25, 36, 49, …) that divides it evenly.
- \(\color{blue}{\sqrt{50} = \sqrt{(25 \times 2)}}\) (25 is the largest perfect square factor of 50)
Step 2: Apply the Product Property of Radicals
\(\color{blue}{\sqrt{(a \times b)} = \sqrt{a} \times \sqrt{b}}\)
- \(\color{blue}{\sqrt{50} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2}}\)
Step 3: Simplify the Perfect Square Root
Take the square root of the perfect square factor. Leave the remaining factor under the radical.
Variables Under the Radical
For even powers, the square root simply halves the exponent: \(\color{blue}{\sqrt{(x^{8})} = x^{4}}\). For odd powers, pull out the highest even power: \(\color{blue}{\sqrt{(x^{5})} = x^{2}\sqrt{x}}\).
Step-by-Step Summary
- Find the largest perfect square factor of the radicand.
- Rewrite the radicand as a product of the perfect square and the remaining factor.
- Split into two radicals using the product property.
- Take the square root of the perfect square factor.
- Write the simplified form: whole-number coefficient × √(remaining factor).
Watch: Simplifying Radicals (Concept Lesson)
Khan Academy explains the concept and product property clearly with multiple examples:
Simplifying Radical Expressions — Worked Examples
Example 1: Simplify \(\color{blue}{\sqrt{50}}\).
Largest perfect square factor: \(\color{blue}{25}\). Rewrite: \(\color{blue}{\sqrt{(25 \times 2)} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2}}\).
Example 2: Simplify \(\color{blue}{\sqrt{72}}\).
Largest perfect square factor: \(\color{blue}{36}\). Rewrite: \(\color{blue}{\sqrt{(36 \times 2)} = 6\sqrt{2}}\).
Example 3: Simplify \(\color{blue}{\sqrt{48}}\).
Largest perfect square factor: \(\color{blue}{16}\). Rewrite: \(\color{blue}{\sqrt{(16 \times 3)} = 4\sqrt{3}}\).
Example 4: Simplify \(\color{blue}{\sqrt{(18x^{4})}}\) (\(\color{blue}{x \ge 0}\)).
Factor: \(\color{blue}{\sqrt{(9 \times 2 \times x^{4})} = \sqrt{9} \times \sqrt{(x^{4})} \times \sqrt{2} = 3x^{2}\sqrt{2}}\).
More Practice: Simplifying Square Roots Video
This Khan Academy video provides additional step-by-step simplification examples:
Exercises for Simplifying Radical Expressions
Simplify each radical expression.
- \(\color{blue}{\sqrt{12}}\)
- \(\color{blue}{\sqrt{45}}\)
- \(\color{blue}{\sqrt{98}}\)
- \(\color{blue}{\sqrt{200}}\)
- \(\color{blue}{\sqrt{27}}\)
- \(\color{blue}{\sqrt{80}}\)
Answers
- \(\color{blue}{2\sqrt{3}}\)
- \(\color{blue}{3\sqrt{5}}\)
- \(\color{blue}{7\sqrt{2}}\)
- \(\color{blue}{10\sqrt{2}}\)
- \(\color{blue}{3\sqrt{3}}\)
- \(\color{blue}{4\sqrt{5}}\)
Want More Practice?
We haven’t published a worksheet built specifically for Simplifying Radical Expressions just yet. In the meantime, the free worksheets below cover closely related skills and concepts. If you’d like extra practice, download any that look helpful, complete the problems, and check your work — they’re a great way to reinforce what you learned on this page and strengthen the foundations this topic builds on:
- Download Properties of Exponents Worksheet
- Download Graphing Square Root Functions Worksheet
- Download Rational and Irrational Numbers Worksheet
Frequently Asked Questions
What is the product property of radicals?
The product property states \(\color{blue}{\sqrt{(\text{ ab })} = \sqrt{a} \times \sqrt{b}}\) for non-negative values. This lets you split a radical into two parts: one that is a perfect square and one that remains under the radical.
How do I know which perfect square factor to use?
Use the largest perfect square factor. For example, \(\color{blue}{\sqrt{72}}\) could be split as \(\color{blue}{\sqrt{(4 \times 18)}}\), giving \(\color{blue}{2\sqrt{18}}\), which still needs simplifying. Using \(\color{blue}{\sqrt{(36 \times 2)} = 6\sqrt{2}}\) gets you all the way to simplest form in one step.
Is √2 in simplest form?
Yes. \(\color{blue}{\sqrt{2}}\) cannot be simplified further because 2 has no perfect-square factors other than 1.
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