Radical Calculator
Radical calculator — evaluate ⁿ√R for any index with decimal value and checks. Free online radical calculator with visuals, practice and study plan.
Name the parts — index, radicand, root — and the decimal follows: √50 ≈ 7.071.
- Grade level
- 8–12
- Curriculum band
- Difficulty
- Intermediate
- For this topic
- Time to master
- 2 weeks
- With daily practice
- Tools on page
- 7 on page
- Calculator + practice
Calculator
Radical — radicand R and index n
Radical Summary
Breakdown
| Property | Value |
|---|---|
| R^(1/n) form | 7.071067811865 |
| valueⁿ check | 50 |
| Root name | Square root |
| √R (n=2 contrast) | 7.071067811865 |
Results
Radical value (ⁿ√R)
7.071067811865
Decimal value of the radical
50
Value inside the radical
2
Which root was taken
50
Raising the value returns R
Results follow real-number radical rules. The index must be an integer ≥ 2, and even indices require R ≥ 0.
Statistics
Live readouts from your current inputs — the anatomy, bars and cards all move together.
Answer ⁿ√R
7.071067811865
n = 2
Radicand R
50
Inside the radical
Check valueⁿ
50
Must equal R
Index
2
Square root
Visualizations
R vs ⁿ√R
Value across indices (R = current)
Magnitude
Formula
A radical is root notation: the index picks the root, the radicand is the target, the value is the answer.
Alternative Form
Variables
| Symbol | Name | Unit | Description |
|---|---|---|---|
| R | Radicand | number | Value inside the radical symbol |
| n | Index | integer | Integer ≥ 2 selecting the root |
| ⁿ√R | Root value | number | Decimal value with (ⁿ√R)ⁿ = R |
Explanation
√50 ≈ 7.071 sits between √49 = 7 and √64 = 8. Raising the index compresses the value toward 1.
Derivation
ⁿ√R = R^(1/n) since (R^(1/n))ⁿ = R. For R < 0 with odd n, ⁿ√R = −ⁿ√|R|.
Step-by-Step Calculation
Follow each step to understand how the result is calculated from your inputs.
Name the parts: index n and radicand R.
Validate: integer index ≥ 2; even n needs R ≥ 0.
Evaluate as a fractional power.
Verify by raising to the index.
Confirm the check matches R.
State the final equality.
Worked Examples
See how the calculator works with different scenarios.
Scenario
Evaluate the square root of 50.
Inputs
Calculation
√50 ≈ 7.0710679 (also 5√2)
Result
≈ 7.071
Interpretation
Between √49 = 7 and √64 = 8, just above 7.
Understanding Your Result
Your Result: Radical value (ⁿ√R): 7.071067811865
Typical range: √50≈7.071, ∛(−27)=−3, ∜81=3; R = 0 gives 0 for every index.
What This Means
The result is the decimal value of ⁿ√R — the number that, raised to n, returns R.
Why It Matters
Radicals appear in geometry diagonals, standard deviation, physics half-powers, and tolerances.
Professional Interpretation
Report the decimal here; move to the simplifying page when exact a√b form is required.
Recommendations
- •Verify with valueⁿ = R
- •Name parts: index, radicand, root
- •Check parity before negative radicands
- •Simplify perfect factors separately
Learn — The Complete Guide
A radical ⁿ√R asks which number raised to n gives R. This calculator evaluates the decimal value with guards.
The core concept
Index n selects the root; radicand R is the target. ⁿ√R = R^(1/n) unifies radicals with exponents.
- A radical names three things: index, radicand, root value
- ⁿ√R = R^(1/n) unifies radicals with exponents
- Raising the value to n (valueⁿ = R) proves it
Derivations — why the formulas work
Radical as fractional power
- 1Let y = ⁿ√R, so yⁿ = R
- 2Raise to 1/n: y = R^(1/n) ✓
Verification identity
- 1Let y = ⁿ√R
- 2By definition yⁿ = R ✓
Where you meet this in real life
Learn this first — prerequisites in order
- 1Square roots (√144 = 12) fluently
- 2Powers: yⁿ = R forward direction
Careers that use this daily
- •Photographer — √2 f-stop sequence
- •Architect — diagonal and paper ratios
- •Data analyst — standard deviation
- •Physicist — square-root scaling laws
Why it helps
- •Decimal value with verification
- •Vocabulary-first radical anatomy
- •Explicit parity guards
Best practices
- •Check answers with valueⁿ = R
- •Estimate with nearby perfect powers
- •Confirm n is an integer ≥ 2
Limitations — what this tool does not do
- •Decimal output — exact a√b form is on the simplifying page
- •Integer indices ≥ 2 only
- •Real numbers only
Curriculum map — where this topic lives in school
| Level | Grades | Standards | Skills |
|---|---|---|---|
| Middle School | 7–8 | 8.EE | √ and ∛ vocabulary · Estimation · Perfect powers |
| Algebra I | 8–9 | HSN-RN | ⁿ√R notation · Fractional powers · Domain rules |
| Geometry | 9–10 | HSG-SRT | Diagonal radicals · Distance formula · Exact vs decimal |
Timeline — how this idea grew up
1525
Rudolff prints √
Die Coss introduces the radical sign.
1637
Descartes adds the bar
The vinculum fixes grouping over the radicand.
1748
Euler’s power link
ⁿ√R = R^(1/n) merges radicals into exponents.
A little history
- • The √ sign debuted in a 1525 German algebra book.
- • Euler recast every radical as a fractional power.
Exam-day checklist
- • Label index and radicand before computing.
- • Verify: valueⁿ must equal R.
- • Keep full precision until the final step.
Myths vs facts
Practical Tips
- •Index 2 hides — √50 means ⁿ√50 with n = 2
- •Raise the value to n to verify
- •Even n needs R ≥ 0
- •Need 5√2 form? Visit simplifying radicals
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 radical?
What does √50 equal as a decimal?
What is the index of a radical?
Why do even indices reject negative radicands?
How do you verify a radical answer?
How do I use this radical 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.
- Radical
- The ⁿ√ symbol plus its radicand.
- Radicand
- The value inside the radical.
- Index
- The small n selecting which root.
- Principal root
- The non-negative even-index root.
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.
√ hides its index
The square root is the only radical allowed to drop the 2 — √50 means ²√50.
Rudolff printed √ in 1525
Die Coss introduced the radical sign; Descartes added the overline later.
√50 = 5√2 exactly
Factoring out 25 leaves √2 — the ratio behind A4 paper (√2:1).
Radicals root phone screens
A “6.1-inch display” is Pythagoras: width² + height² rooted.
Compare Side by Side
Same radicand, three indices — higher n compresses toward 1.
| Index | R = 50 | Says | Value |
|---|---|---|---|
| n = 2 (√) | Square root | 7.071067811865 | |
| n = 3 (∛) | Cube root | 3.68403149864 | |
| n = 4 (∜) | Fourth root | 2.659147948472 |
Recent Calculations
Your last runs on this page — stored only in this browser, never uploaded.
ⁿ√50 (n=2) = 7.071067811865
Current inputs — live
Study Plan
A week-by-week path from prerequisites to mastery. Your checklist progress saves in this browser.
Prerequisites
- ▸Square and cube roots
- ▸Vocabulary: index, radicand
Mastery progress — 0%
- I can name every part of ⁿ√R
- I can evaluate radicals for n = 2…5
- I can apply the parity domain rule
- I can verify with valueⁿ = R
- 1
Week 1 · ~15 min/day
Radical vocabulary
Goals
- Name index, radicand, and value
- Evaluate √ and ∛ of perfect powers
Practice
Label 10 radicals with all three parts before evaluating.
- 2
Week 2 · ~20 min/day
Decimal evaluation
Goals
- Convert ⁿ√R to R^(1/n)
- Verify with valueⁿ = R
Practice
Estimate 5 radicals by bracketing, then confirm here.
- 3
Week 3 · ~20 min/day
Exact vs decimal
Goals
- Decide when 5√2 beats 7.071
- Apply radicals in Pythagoras
Practice
Solve one diagonal problem both ways: decimal and exact.
Exam tips
- •Write the index explicitly (except √).
- •Raise the value to n 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.
Evaluate √49.
Name the index and radicand of ∛(−27).
Estimate √50 to one decimal.
Is ∜(−16) real?
A square has area 50 cm². 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 |
|---|---|---|
| ⁿ√R = R^(1/n) | Radical as power | √50 = 50^0.5 |
| (ⁿ√R)ⁿ = R | Verification identity | 7.071² ≈ 50 |
| √ = ²√ | Hidden index 2 | √50 = ²√50 |
| Even n ⟹ R ≥ 0 | Domain guard | ∜(−16) warns |
| √(a²b) = a√b | Preview of simplifying | √50 = 5√2 |
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Page Information
2026-09-29
3.1.0-math
Free to use for educational and professional purposes.