Square Root Calculator
Square root calculator — compute √x with a squaring check and negative-input guard. Free online sqrt calculator with visuals, practice and study plan.
Undo any square: the root of 144 walks straight back to 12.
- Grade level
- 7–11
- Curriculum band
- Difficulty
- Beginner
- For this topic
- Time to master
- 1 week
- With daily practice
- Tools on page
- 6 on page
- Calculator + practice
Calculator
Square Root — x must be ≥ 0
Square Root Summary
Breakdown
| Property | Value |
|---|---|
| Domain: x ≥ 0? | Yes ✓ |
| √x (principal) | 12 |
| −√x (twin root) | -12 |
| root² check | 144 |
Results
Square root (√x)
12
Principal (non-negative) root
144
The radicand you rooted
144
Squaring the answer returns x
-12
The second solution of t² = x
Results follow real-number root rules. The radicand must be ≥ 0 — negative inputs show a warning instead of a value.
Statistics
Live readouts from your current input — the bars, curve and cards all move together.
Answer √x
12
√144
Input x
144
Radicand
Check root²
144
Must equal x
−√x
-12
Twin root
Visualizations
x vs √x
Roots near your input
Root size
Formula
The square root inverts squaring for non-negative inputs. Squaring the result must return the radicand.
Alternative Form
Variables
| Symbol | Name | Unit | Description |
|---|---|---|---|
| x | Radicand | number | Value under the radical; must be ≥ 0 |
| √x | Principal root | number | Non-negative value with (√x)² = x |
| −√x | Twin root | number | Second solution of t² = x |
Explanation
Roots compress quadratic growth back to linear: √144 = 12. The curve rises steeply near 0 then flattens.
Derivation
If y = √x then y² = x, y ≥ 0. Product rule: (√a·√b)² = ab, so √(ab) = √a·√b.
Step-by-Step Calculation
Follow each step to understand how the result is calculated from your inputs.
Read the radicand.
Confirm the domain: radicand ≥ 0.
Take the principal (non-negative) root.
Verify by squaring the answer.
Note the twin root: −√x also squares to x.
State the final equality.
Worked Examples
See how the calculator works with different scenarios.
Scenario
Compute √144.
Inputs
Calculation
√144 = 12 since 12² = 144
Result
12
Interpretation
A 144-tile square has 12-tile sides.
Understanding Your Result
Your Result: Square root (√x): 12
Typical range: √100=10, √144=12, √2≈1.414; roots of numbers < 1 exceed the input.
What This Means
The result is the side length of a square with area x — the inverse of squaring.
Why It Matters
Roots power Pythagoras, standard deviation, the quadratic formula, and distance in every dimension.
Professional Interpretation
Always return the principal root; list ± when solving t² = x equations.
Recommendations
- •Verify with root² = x
- •Bracket between perfect squares first
- •Simplify: √50 = 5√2
- •Reject negative radicands in real work
Learn — The Complete Guide
The square root undoes squaring for x ≥ 0. This calculator computes √x with an automatic squaring check.
The core concept
√x is the non-negative y with y² = x. It mirrors squaring: one compresses what the other expands.
- √x is the non-negative number squaring to x
- Real roots require x ≥ 0 — negatives warn, not compute
- Squaring the answer (root² = x) proves every result
Derivations — why the formulas work
Verification identity
- 1Let y = √x
- 2By definition y ≥ 0, y² = x
- 3So (√x)² = x ✓
Product rule
- 1(√a·√b)² = (√a)²(√b)²
- 2= a·b
- 3So √(ab) = √a·√b ✓
Where you meet this in real life
Learn this first — prerequisites in order
- 1Squaring fluently, including 12² = 144
- 2Perfect squares to at least 15²
Careers that use this daily
- •Civil engineer — hypotenuse and load geometry
- •Statistician — standard deviation daily
- •Carpenter — square corners via 3-4-5 roots
- •Data scientist — distance metrics with √
Why it helps
- •Instant root with verification
- •Explicit negative-input warning
- •Shows the twin ± root
Best practices
- •Check answers by squaring
- •Estimate with perfect-square brackets
- •Include ± when solving equations
Limitations — what this tool does not do
- •Real numbers only — negatives rejected
- •Display rounds to 12 significant digits
- •Irrational roots shown as decimals
Curriculum map — where this topic lives in school
| Level | Grades | Standards | Skills |
|---|---|---|---|
| Middle School | 7–8 | 8.EE | Perfect-square roots · Estimation · Irrationals intro |
| Algebra I | 8–9 | HSN-RN | Radical rules · t² = x equations · Simplification |
| Geometry | 9–10 | HSG-SRT | Pythagoras · Distance formula · Root checks |
Timeline — how this idea grew up
c. 1800 BCE
Babylonian √2 tablet
YBC 7289 records √2 for diagonal calculations.
c. 500 BCE
√2 proved irrational
Hippasus shows no fraction squares to 2.
1525
The √ symbol appears
Christoff Rudolff prints the first radical sign.
A little history
- • Babylonians computed √2 to 6-digit accuracy 3800 years ago.
- • The √ symbol debuted in print in 1525 Germany.
Exam-day checklist
- • Check x ≥ 0 before touching the calculator.
- • Bracket non-perfect roots between squares.
- • Add ± when the equation is t² = x.
Myths vs facts
Practical Tips
- •Only x ≥ 0 has a real root
- •Square the answer to verify
- •Bracket: √50 sits between 7 and 8
- •Use ± when solving t² = x
Common Mistakes
Assumptions
This calculator makes the following assumptions to provide estimates:
Limitations
Be aware of these limitations when using this calculator:
Frequently Asked Questions
Answers to common questions about this calculator.
What is a square root?
How do you verify √144 = 12?
Can you take the square root of a negative?
What is the square root of 0?
How do you estimate √50 mentally?
How do I use this square root calculator?
Related Calculators
Explore more calculators related to this topic.
Related Guides
Deepen your understanding with these educational resources.
Glossary
Technical terms used in mortgage calculations explained in plain language.
- Radicand
- The value under the √ symbol.
- Principal root
- The non-negative root denoted by √.
- Perfect square
- An integer with an integer root (144 = 12²).
- Twin root
- −√x, the second solution of t² = x.
Authoritative References
Trusted sources and standards referenced by this calculator.
Applicable Standards
Diagrams & Illustrations
Visual intuition for every core idea. Select a card to enlarge it.
Did You Know?
Short intuition builders — great for class warm-ups.
√2 sparked a crisis
Hippasus proved √2 is irrational — legend says the Pythagoreans drowned him at sea.
Babylonians nailed √2
The YBC 7289 tablet gives √2 to six sexagesimal digits — accurate to 1 in 2 million.
√144 = 12 is a gross root
144 is a dozen dozen, so root 12 anchors cartons, inches, and clock faces.
Roots slow down traffic math
Braking distance grows with speed squared — stopping math is all square roots.
Compare Side by Side
Three ways to root the same input — only the square root refuses negatives.
| Root | x = 144 | Rule | Value |
|---|---|---|---|
| Square root (√x) | Needs x ≥ 0 | 12 | |
| Cube root (∛x) | Accepts negatives | 5.241482788418 | |
| Half power (x^0.5) | Same as √x | 12 |
Recent Calculations
Your last runs on this page — stored only in this browser, never uploaded.
√144 = 12
Current inputs — live
Study Plan
A week-by-week path from prerequisites to mastery. Your checklist progress saves in this browser.
Prerequisites
- ▸Squaring numbers (x²)
- ▸Perfect squares to 15²
Mastery progress — 0%
- I can root perfect squares to 225 instantly
- I can explain why negatives have no real root
- I can bracket-estimate any root
- I can verify with root² = x
- 1
Week 1 · ~15 min/day
Perfect-square roots
Goals
- Recall √1–√225 for perfect squares
- State the x ≥ 0 domain rule
Practice
Drill 10 perfect-square roots daily; square each answer aloud.
- 2
Week 2 · ~20 min/day
Estimating non-perfect roots
Goals
- Bracket roots between perfect squares
- Simplify √50 = 5√2 style radicals
Practice
Estimate 5 non-perfect roots by bracketing, then confirm here.
- 3
Week 3 · ~20 min/day
Roots in geometry
Goals
- Apply √ to square areas
- Include ± when solving t² = x
Practice
Solve one hypotenuse problem end-to-end with a root check.
Exam tips
- •Check the domain first: x ≥ 0 or stop.
- •Square the answer to confirm.
Practice Problems
0/5 solvedTry each problem first, peek at the hint only if stuck, then reveal the solution and mark it solved.
Compute √81.
Compute √0.
Estimate √50 to one decimal.
Solve t² = 144.
A square rug covers 144 ft². What is its side?
Printable Worksheet
Generate a fresh problem set for class or homework. Print it, or copy the answer key separately.
- 1.18 × 47 =
- 2.8 ÷ 11 =
- 3.38 × 26 =
- 4.35 ÷ 4 =
- 5.21 − 16 =
- 6.2 + 26 =
- 7.14 ÷ 12 =
- 8.23 ÷ 3 =
- 9.49 ÷ 7 =
- 10.22 − 17 =
- 11.41 + 18 =
- 12.47 − 35 =
Formula Cheat Sheet
Every formula on this page in one printable table.
| Formula | Meaning | Example |
|---|---|---|
| √x = y ⟺ y² = x, y ≥ 0 | Definition of principal root | √144 = 12 |
| (√x)² = x | Verification identity | 12² = 144 |
| √(ab) = √a·√b | Product rule (a,b ≥ 0) | √50 = 5√2 |
| t² = x ⟹ t = ±√x | Twin roots in equations | t²=144 → ±12 |
| √0 = 0 · √1 = 1 | Anchor values | Domain starts at 0 |
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Page Information
2026-09-29
3.1.0-math
Free to use for educational and professional purposes.