Simplify Radical Expressions Calculator
Simplify square roots, cube roots, and nth roots instantly with full step-by-step solutions.
| Property | Value |
|---|
| Original Radical | Simplified Form | Decimal |
|---|---|---|
| √8 | 2√2 | 2.8284… |
| √12 | 2√3 | 3.4641… |
| √18 | 3√2 | 4.2426… |
| √20 | 2√5 | 4.4721… |
| √24 | 2√6 | 4.8990… |
| √27 | 3√3 | 5.1962… |
| √32 | 4√2 | 5.6569… |
| √45 | 3√5 | 6.7082… |
| √48 | 4√3 | 6.9282… |
| √50 | 5√2 | 7.0711… |
| √72 | 6√2 | 8.4853… |
| √75 | 5√3 | 8.6603… |
| ∛8 | 2 | 2.0000 |
| ∛24 | 2∛3 | 2.8845… |
| ∛54 | 3∛2 | 3.7798… |
How to Simplify Radical Expressions
Simplifying radical expressions means rewriting them in their simplest form. A simplified radical expression has no perfect power factors remaining inside the radical sign. The process involves factoring the radicand and extracting any perfect nth powers.
Steps to Simplify a Square Root
- Find the prime factorization of the number inside the radical
- Group the prime factors into pairs (for square roots)
- Move each pair out of the radical as a single factor
- Multiply all factors outside the radical together
- Multiply any remaining factors inside the radical together
Example: To simplify the square root of 72, factor 72 = 2 × 2 × 2 × 3 × 3. Group pairs: (2 × 2) and (3 × 3) come out, leaving 2 inside. Result: 6 times the square root of 2.
Simplifying Cube Roots
For cube roots, group prime factors into sets of three instead of pairs. Example: the cube root of 54 factors as 2 × 3 × 3 × 3. The group of three 3s comes out as 3, leaving 2 inside. Result: 3 times the cube root of 2.
Rules for Simplified Radical Form
A radical expression is considered fully simplified when all of these conditions are met:
- The radicand contains no perfect square factors (for square roots) or perfect nth power factors
- There are no fractions inside the radical sign
- There are no radicals in the denominator of any fraction (rationalized form)
- The index of the radical and the exponent of the radicand share no common factors
Perfect Squares and Perfect Cubes
Knowing common perfect squares and cubes makes simplification faster:
- Perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225
- Perfect cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000
- Perfect 4th powers: 1, 16, 81, 256, 625
Frequently Asked Questions
To simplify a radical expression, find the largest perfect power that divides the radicand. Extract its root outside the radical and leave the remaining factor inside. For example, the square root of 72 simplifies to 6 times the square root of 2, because 72 = 36 × 2 and the square root of 36 is 6.
A radical expression is fully simplified when the radicand has no perfect square factors (for square roots), no perfect cube factors (for cube roots), no fractions under the radical, and no radicals in the denominator.
To simplify a square root, factor the number inside the radical into a perfect square and a remaining factor. Write the square root of the perfect square outside the radical and leave the remaining factor inside. For example, the square root of 50 = the square root of (25 × 2) = 5 times the square root of 2.
Perfect squares are numbers that result from squaring an integer: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, and so on.
No. If the radicand is a prime number, the radical is already in its simplest form. For example, the square root of 7 cannot be simplified further.