Simplifying Radical Expressions: Square Roots Made Easy

 

Math Made Friendly

Simplifying Radical Expressions: Square Roots and Beyond

Learn how to find perfect-square factors, simplify roots, and combine like radicals.

The Big Idea

A radical expression contains a root, such as √36 or √50. To simplify a square root, look for the largest perfect-square factor inside the radicand, the number under the radical sign. Pull the square root of that perfect square outside, then leave any factor that is not a perfect square inside.

The Formula to Remember

√(a × b) = √a × √b

Choose a to be the largest perfect-square factor you can find. For example, 50 = 25 × 2, so √50 = √(25 × 2) = √25 × √2 = 5√2.

Memory tip: Find the square, split the root, pull it out, and check.

Step-by-Step Method

1. Factor the radicand into a perfect square times another factor.
2. Use the product rule to separate the roots.
3. Take the square root of the perfect square.
4. Leave the remaining factor under the radical.
5. Check that no perfect-square factor greater than 1 is still hiding inside.

Worked Examples

Example 1: Simplify √72

The largest perfect-square factor of 72 is 36.

√72 = √(36 × 2) = √36 × √2 = 6√2

Example 2: Simplify √180

Use the largest perfect-square factor, 36.

√180 = √(36 × 5) = √36 × √5 = 6√5

Example 3: Simplify 3√48

First simplify the radical. Since 48 = 16 × 3, √48 = 4√3. Then multiply by the coefficient 3.

3√48 = 3(4√3) = 12√3

Example 4: Add like radicals

Radicals can be combined only when their radical parts match. Add the coefficients and keep the common radical.

2√5 + 7√5 = 9√5

Example 5: Simplify a cube root

The same idea works beyond square roots. For cube roots, look for perfect cubes such as 8, 27, or 125.

cube root of 54 = cube root of (27 × 2) = 3 × cube root of 2

A Quick Reference Table

On a small phone screen, swipe horizontally inside the table if needed.

Expression	Helpful factorization	Simplified form
√12	12 = 4 × 3	2√3
√75	75 = 25 × 3	5√3
√200	200 = 100 × 2	10√2
cube root of 40	40 = 8 × 5	2 × cube root of 5

Common Mistakes to Avoid

• Stopping too soon: √48 = 4√3, not √48 or 2√12. Keep simplifying until no perfect-square factor remains.
• Adding unlike radicals: √2 + √3 cannot be combined into √5. Only matching radical parts combine.
• Adding inside the radical: √a + √b is generally not equal to √(a + b).
• Forgetting the coefficient: In 3√20, the 3 is outside the radical and must be multiplied after simplifying √20.
• Ignoring the principal-root convention: √25 means the principal, nonnegative square root, so √25 = 5.

Practice Questions

Try these before opening the answer section. On a small phone screen, swipe horizontally inside an equation box if needed.

1. Simplify √20.
2. Simplify √98.
3. Simplify 4√45.
4. Simplify 2√18 + √8.
5. Can √6 + √24 be simplified and combined? Show your steps.
6. Simplify the cube root of 81.

Answer Section

Final Takeaway

To simplify a radical, find a perfect-power factor, split the radical, pull out the exact root, and combine only like radicals. The same strategy works with square roots, cube roots, and other roots.