How to Simplify Radical Expressions? (+FREE Worksheet!)

Simplify square-root expressions by identifying perfect-square factors.

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:

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• 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

1. Find the largest perfect square factor of the radicand.
2. Rewrite the radicand as a product of the perfect square and the remaining factor.
3. Split into two radicals using the product property.
4. Take the square root of the perfect square factor.
5. 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.

1. \(\color{blue}{\sqrt{12}}\)
2. \(\color{blue}{\sqrt{45}}\)
3. \(\color{blue}{\sqrt{98}}\)
4. \(\color{blue}{\sqrt{200}}\)
5. \(\color{blue}{\sqrt{27}}\)
6. \(\color{blue}{\sqrt{80}}\)

Answers

1. \(\color{blue}{2\sqrt{3}}\)
2. \(\color{blue}{3\sqrt{5}}\)
3. \(\color{blue}{7\sqrt{2}}\)
4. \(\color{blue}{10\sqrt{2}}\)
5. \(\color{blue}{3\sqrt{3}}\)
6. \(\color{blue}{4\sqrt{5}}\)

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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 a fully simplified radical expression?

A radical expression is fully simplified when three things are true: no perfect-square factor (or perfect cube, for cube roots) is left under the radical, no fraction is under the radical, and no radical is left in a denominator. So √20 is not simplified, but 2√5 is. Likewise, 1/√3 should be rewritten as √3/3.

How do you simplify radical expressions step by step?

Find the largest perfect square that divides the radicand, split the radical, and take its root. For √180: the perfect squares 4, 9 and 36 all divide 180, and 36 is the largest. Write √180 = √(36 × 5) = √36 × √5 = 6√5. If you start with a smaller square, like 4, keep going until no perfect square is left.

How do you simplify radical expressions with variables and exponents?

Split each variable power into the largest even power times what is left, then take square roots of the even parts by halving their exponents. For √(50x⁵y²), write it as √(25x⁴y² × 2x). The square root of 25x⁴y² is 5x²y, so the answer is 5x²y√(2x), assuming x and y are nonnegative.

How do you simplify radical expressions with a number in front?

Simplify the radical first, then multiply what comes out by the number already in front. For 3√48, write 48 as 16 × 3, so √48 = 4√3. Then 3 × 4√3 = 12√3. The number in front is a coefficient, so it multiplies whatever you pull out of the radical; it never goes under the root sign.

How do you simplify radical expressions with cube roots?

Look for perfect-cube factors such as 8, 27, 64 and 125, and for variable exponents that are multiples of 3. For ∛54, write 54 = 27 × 2, so ∛54 = 3∛2. For ∛(16x⁴), write 16x⁴ = 8x³ × 2x, which gives 2x∛(2x). Cube roots of negative numbers are allowed: ∛(−8) = −2.

How do you simplify radical expressions by rationalizing the denominator?

Multiply the top and bottom by the radical in the denominator so it becomes a whole number. For 6/√3, multiply by √3/√3 to get 6√3/3, which reduces to 2√3. For a denominator like 2 + √3, multiply by its conjugate, 2 − √3, because (2 + √3)(2 − √3) = 4 − 3 = 1.

Related Topics

• Adding and Subtracting Radical Expressions
• Multiplying Radical Expressions
• Rationalizing Radical Expressions
• Radical Equations
• Domain and Range of Radical Functions