A "simplified radical" is a square root rewritten so that no perfect-square factors remain under the radical sign. The simplified form is cleaner, easier to compare, and the form your math teacher expects on the answer key. Here's the full procedure.
The core rule
√(a × b) = √a × √b
That single identity is everything. The trick is choosing a and b so that one of them is a perfect square.
Example: √72.
- Factor 72: 72 = 36 × 2.
- √72 = √36 × √2 = 6√2.
- Done. 6√2 is the simplified radical form.
Step by step
- Find the largest perfect-square factor of the radicand (the number under the root).
- Write the radicand as that perfect square × the remaining factor.
- Take the square root of the perfect square; leave the rest under the radical.
If the largest perfect-square factor isn't obvious, use prime factorization:
Example: √200.
- Prime factor: 200 = 2³ × 5² = 2 × 2² × 5².
- Perfect square parts: 2² × 5² = 4 × 25 = 100.
- 200 = 100 × 2, so √200 = √100 × √2 = 10√2.
Spotting perfect-square factors fast
Memorize the first few perfect squares: 4, 9, 16, 25, 36, 49, 64, 81, 100. Run through them mentally:
- √48: divisible by 4? Yes. 48 / 4 = 12. So √48 = 2√12. But 12 has another factor of 4, so √12 = 2√3. Final: √48 = 4√3.
- √98: divisible by 49? Yes. 98 / 49 = 2. So √98 = 7√2.
- √300: divisible by 100? Yes. 300 / 100 = 3. So √300 = 10√3.
The trick: don't stop at the first perfect-square factor — keep simplifying until no perfect squares remain.
Variables under radicals
Same rules apply: √(x²) = |x| (absolute value because square roots are nonnegative). For positive variables, just write x.
Examples:
- √(x⁴) = x²
- √(x⁵) = √(x⁴ × x) = x²√x
- √(50x³) = √(25 × 2 × x² × x) = 5x√(2x)
The general rule: even powers come out cleanly (divide the exponent by 2). Odd powers leave one factor under the radical.
Adding and subtracting radicals
You can only combine "like radicals" — radicals with the same number under the root sign:
- 3√2 + 5√2 = 8√2 (like terms)
- 3√2 + 5√3 (cannot be combined)
- √50 + √8 = 5√2 + 2√2 = 7√2 (after simplifying)
Always simplify first — sometimes terms that look unlike will combine after simplification.
Multiplying radicals
Multiply the coefficients, multiply the radicands, then simplify:
- (2√3)(5√6) = 10√18 = 10 × 3√2 = 30√2
- (√x)(√y) = √(xy)
Rationalizing the denominator
U.S. math convention: the answer should not have a radical in the denominator. Multiply top and bottom by whatever clears it.
Example: 1/√2.
Multiply by √2/√2: 1/√2 × √2/√2 = √2/2. The result is mathematically equal, but with a clean denominator.
For binomial denominators (a + √b), multiply by the conjugate (a − √b) to use the difference-of-squares pattern.
When to leave it as a decimal vs simplify
Algebra and geometry teachers want simplified radicals. Physics and engineering happily use decimals — 7.071 is more useful than 5√2 when plugging into a formula. Both are correct mathematically; they're just different conventions.
Cube roots and higher roots
The same simplification idea applies to cube roots and beyond. Pull out perfect cubes (8, 27, 64, 125, 216, 343, 512, 729, 1000) for cube roots; pull out fourth powers (16, 81, 256, 625) for fourth roots.
- ∛54 = ∛(27 × 2) = 3∛2
- ∛250 = ∛(125 × 2) = 5∛2
- ⁴√48 = ⁴√(16 × 3) = 2 × ⁴√3
The general rule for nth roots: pull out factors that are perfect nth powers. The exponent of any factor under the root divides cleanly by n, with the remainder staying inside.
Adding fraction radicals
Fraction radicals follow the same simplification logic, with one extra step — rationalize the denominator at the end.
Example: simplify 3/√50.
- Simplify the radical: √50 = 5√2.
- Substitute: 3/(5√2).
- Rationalize: multiply numerator and denominator by √2 → (3√2)/(5 × 2) = (3√2)/10.
Practice this on textbook problems: the simplification step often makes the final rationalization much easier.
Verify quickly
Our square root calculator shows both the decimal and the simplified radical form for any whole-number radicand. Useful for checking homework — if the calculator says √72 = 6√2 and you wrote 4√3, you know to double-check the factoring.