Simplify radicals calculator
Enter a number under the root, get the simplest form.
How do I simplify a square root?
Find the largest perfect square that divides the number and take its root outside: √72 = √(36 × 2) = 6√2 ≈ 8.485.
How do I simplify a cube root?
The same way with perfect cubes: ∛54 = ∛(27 × 2) = 3∛2.
When is a radical in simplest form?
When the number under the root has no factor that is a perfect square (or cube, for cube roots) other than 1 — √30 cannot be simplified.
Can I take the cube root of a negative number?
Yes — odd roots of negatives are real: ∛−54 = −3∛2. Even roots of negative numbers are not real numbers.
How does the calculator find the factors?
It splits the number into prime factors and moves each complete group of 2 (or 3, and so on) outside: 72 = 2³ × 3², so √72 = 2 × 3 × √2.
How do I simplify a fourth root or a higher root?
Look for perfect fourth (fifth, …) powers, not squares. 243 = 81 × 3 and 81 = 3⁴, so ⁴√243 = 3 × ⁴√3 ≈ 3.9482 (OpenStax Intermediate Algebra 2e, section 8.2). The 9 in 243 = 9 × 27 is a perfect square but not a fourth power, so it cannot come out.
Can I split √(a + b) into √a + √b?
No. The product property ⁿ√(ab) = ⁿ√a × ⁿ√b works only for multiplication. √(9 + 16) = √25 = 5, but √9 + √16 = 3 + 4 = 7.
How do I simplify the root of a fraction?
Use the quotient property ⁿ√(a ÷ b) = ⁿ√a ÷ ⁿ√b (b ≠ 0) and enter the top and bottom one at a time. For √(45/80): √45 = 3√5 and √80 = 4√5, so the quotient is 3/4, as in the OpenStax example.
The rule and what it assumes
A radical ⁿ√a is in simplest form when the number under it has no factor that is a perfect nth power, where n is the index (OpenStax Intermediate Algebra 2e, section 8.2). The tool is the product property, valid when the roots involved are real. Odd roots of negative numbers are real, as in ∛(−64) = −4, while an even root of a negative number is not a real number (section 8.1), so the calculator says so instead of answering. It accepts whole numbers up to 10¹² and indexes 2 to 10. The decimal beside the radical is rounded; the radical is exact.
Worked example: √500
Factor 500 into primes: 500 = 2² × 5³. The pair of 2s and one pair of 5s leave the root as 2 × 5 = 10, a single 5 stays inside, and √500 = 10√5 ≈ 22.3607. The long way, √(100 × 5) = 10√5, is the answer OpenStax gives.
Common mistakes
Stopping too early: √72 = √(4 × 18) = 2√18 is correct but unfinished, because 18 still contains a 9. Pulling out the largest perfect square, 36, reaches 6√2 ≈ 8.4853 in one step. For a cube root the factor must be a perfect cube such as 8 or 27, not a square. When you simplify by hand with variables, an even root of a square needs absolute value signs (√(x²) = |x|) and an odd root does not; the calculator itself takes numbers only.
Sources: OpenStax, Intermediate Algebra 2e, sections 8.1 “Simplify Expressions with Roots” and 8.2 “Simplify Radical Expressions” (product and quotient properties, definition of simplest form, examples √500, ⁴√243 and √(45/80)). Related tools: square root calculator, factor calculator, exponent calculator and Pythagorean theorem calculator. How we check our calculators.