Square Root Simplifier
Simplify a real square root into a√b, with prime-factor pairs and a decimal check. Distinguish the principal square root from both solutions of x²=n.
YOUR CALCULATION
Result
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Calculation steps
Work through one step at a time or reveal the complete calculation. The steps use your current inputs.
How to calculate
Factor n into primes. Every pair of equal prime factors contributes one factor outside the root; unpaired factors stay inside. The symbol √n means the nonnegative principal root. Solving x²=n also includes its negative when n>0.
√(a²b)=a√b for a≥0; if n>0, x²=n has x=±√n
Worked example
72=2×2×2×3×3=(2×3)²×2. Therefore √72=6√2≈8.48528137. The equation x²=72 has two solutions, +6√2 and −6√2; √72 itself is positive.
Assumptions and input details
This tool accepts one whole radicand from 0 to 1,000,000,000 and works over the real numbers. Negative radicands, fractions and decimals are outside this tool’s input domain. Exact integer factorization determines the simplified radical; decimal values are rounded. Zero has square root zero and x²=0 has one distinct solution. One has square root one and needs no prime factors.
Using this calculator
Can I enter my own problem?
Yes. Replace the example values with your own inputs. The result, calculation steps, tables and applicable diagrams update from those inputs. Blank or invalid values clear the result; example values are never silently substituted for missing inputs.
Are the results exact?
The simplified radical and its integer factors are exact. The decimal value and plotted positions are approximations.
Is my input uploaded?
The calculation runs in your browser. No account, file upload or external AI service is required. Inputs are not saved between page reloads.
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