What Is a Radical?
A radical expression consists of a radical symbol (√), a radicand (the number under the symbol), and optionally a root index (the small number to the upper left of the symbol). When no index is written, a square root (index = 2) is implied. The cube root uses index 3, written as ∛.
Simplifying a radical means extracting perfect power factors from the radicand so the number inside the radical is as small as possible. The simplified form is written as: coefficient × ⁿ√(remaining radicand).
How to Simplify a Square Root
Step 1: Find the prime factorization of the radicand.
Example: 72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3²
Step 2: Group prime factors in pairs (for square roots).
2³ × 3² → one pair of 2s, one pair of 3s, one leftover 2
Step 3: Move one factor from each complete pair outside.
√72 = 2 × 3 × √2 = 6√2
Step 4: Verify: 6² × 2 = 36 × 2 = 72 ✓
Common Simplified Radicals
| Original | Simplified | Decimal |
|---|---|---|
| √8 | 2√2 | 2.828 |
| √12 | 2√3 | 3.464 |
| √18 | 3√2 | 4.243 |
| √20 | 2√5 | 4.472 |
| √50 | 5√2 | 7.071 |
| √72 | 6√2 | 8.485 |
| √75 | 5√3 | 8.660 |
| √98 | 7√2 | 9.899 |
| ∛24 | 2∛3 | 2.884 |
| ∛54 | 3∛2 | 3.780 |