Square Root Calculator
with Step-by-Step AI
Enter any number and get the exact square root, simplified radical form, and a plain-language explanation — instantly and free.
Integers, decimals, fractions, negatives & grade curves supported
Enter any number — integers, decimals, fractions, or negatives all work
Step 2 — Extract the largest perfect square factor:
72 = 36 × 2, where 36 is a perfect square. So √72 = √(36 × 2) = √36 × √2 = 6√2.
Step 3 — Decimal value: 6√2 = 6 × 1.41421… ≈ 8.485
Step 4 — Verify: (6√2)² = 36 × 2 = 72 ✓
Final Answer: √72 = 6√2 ≈ 8.485
From number to explained answer in seconds
Type any number
Enter an integer, decimal, fraction like 9/16, or a negative number. Select Standard, Simplify, or Grade Curve mode.
AI factors and simplifies
The solver finds the prime factorization, extracts perfect square factors, and reduces the radical to simplest form.
Read every step
Each step is labeled and explained — not just a symbol, but the reason behind it. Decimal approximation and verification included.
Radical simplification
Not just a decimal — the calculator returns the fully simplified radical form. √200 becomes 10√2, not just 14.142.
Grade curve calculator
Apply the square root curve to any raw test score. Enter your score, get the curved result, and see why the formula works.
All number types
Integers, decimals, fractions, and negatives (imaginary results). One tool handles every form without switching pages.
Step-by-step, plain English
Factor trees, extraction logic, verification — explained conversationally, not just with notation. Better than a table lookup.
Every kind of square root problem
Square roots solved, explained, simplified
integers, decimals, fractions, negatives
no trial limit ever
full step-by-step solution
MathGPT Free vs. the alternatives
Built for students, teachers, and test-takers
Stuck on simplifying radicals
The step-by-step breakdown shows exactly which perfect square to extract and why — so you learn the method, not just the answer.
Generating worked examples
Pick any number, get a clean worked solution with factor tree included. Copy it into your worksheet or use it to explain the process live.
Checking work before submitting
Verify your radical simplification or exact answer. The tool shows the verification step so you can confirm your reasoning is correct.
Applying the square root curve
Got a raw score of 56 on a curved exam? Enter it in Grade Curve mode: √(56/100) × 100 ≈ 74.8 — a full 18-point boost explained.
Square Root Calculator — Free, Step-by-Step
A square root calculator finds the value that, when multiplied by itself, gives you the original number. For example, the square root of 144 is 12 because 12 × 12 = 144. For non-perfect squares like 72, the result is an irrational number — and this calculator gives you both the simplified radical form (6√2) and the decimal approximation (≈8.485).
Most online calculators return only the decimal. This one shows the full working: prime factorization, perfect square extraction, simplification, and verification. If you need to show your work — or actually understand the process — that matters.
How to Simplify a Square Root
Simplifying a radical means rewriting it in a form where no perfect square remains under the radical sign. The method:
- Factor the number under the radical into its prime factors.
- Identify perfect square pairs in the factorization (pairs of identical primes).
- Pull each pair out from under the radical as a single factor.
- Leave the remaining factors under the radical.
Example: √72 = √(2 × 2 × 2 × 3 × 3) = 2 × 3 × √2 = 6√2. The calculator shows this factor tree automatically.
Square Root Calculator with Steps
A standard calculator gives you √72 = 8.485… and nothing else. This tool shows why: which factors were identified, which perfect squares were extracted, and how the decimal was derived. Each operation is explained in plain English, making it useful for checking homework or studying for a test — not just getting an answer to copy.
If your teacher or course requires showing work, the step-by-step breakdown gives you the logic to write up yourself.
Square Root Curve Calculator for Grades
The square root curve is a common grading adjustment that benefits students with lower raw scores. The formula is:
Curved score = √(raw score ÷ 100) × 100
Examples of how the curve works:
- Raw 64 → √(64/100) × 100 = √0.64 × 100 = 0.8 × 100 = 80
- Raw 49 → √(49/100) × 100 = 0.7 × 100 = 70
- Raw 81 → √(81/100) × 100 = 0.9 × 100 = 90
The Grade Curve mode applies this formula to any score and shows the calculation step by step.
How to Find a Square Root Without a Calculator
For perfect squares, the answer is memorized or derived from the multiplication table. For other numbers, there are two common methods. The prime factorization method works well for simplification: factor the number and pull out perfect square pairs. The estimation and averaging method (also called the Babylonian or Heron’s method) starts with a rough guess, then refines it by averaging the guess with the quotient — repeating until the desired precision is reached.
For exam settings where calculators are not allowed, knowing your perfect squares up to 225 and how to simplify radicals by factoring is usually enough.
Square Root of Fractions and Decimals
For a fraction like 9/16, apply the square root to numerator and denominator separately: √9 / √16 = 3/4. For a decimal like 0.25, convert it to the fraction 1/4 first, then apply the same rule: √(1/4) = 1/2 = 0.5. The calculator handles both forms automatically.
Square Root of Negative Numbers
The square root of a negative number is an imaginary number. By definition, i = √(−1). So √(−9) = √9 × √(−1) = 3i, and √(−50) = 5i√2. The calculator shows the derivation and simplification for any negative input.
Real results from real students
“Every other calculator just gives me the decimal. This one shows the factor tree and why 6√2 is the simplified form. That’s what I actually needed for my test.”
“The grade curve feature saved me so much time. I entered my whole class’s raw scores and could tell them the curved grades instantly — with the calculation shown.”
“I was confused about imaginary numbers. Entering −25 and seeing the step-by-step explanation of why the answer is 5i made more sense than my textbook did.”
Common questions
Your square root is one click away
Type any number. Get the simplified form, decimal value, and every step explained.
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