Nth Root Calculator
Nth root calculator — compute ⁿ√x for any index with domain guards. Free online root calculator with visuals, practice and study plan.
Dial any index: ∜81 = 3, ∛(−27) = −3. Even indices guard negatives; odd wave them through.
- 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
Nth Root — x and index n
Nth Root Summary
Breakdown
| Property | Value |
|---|---|
| x^(1/n) form | 3 |
| rootⁿ check | 81 |
| Parity | Even |
| √x (n=2 contrast) | 9 |
Results
Nth root (ⁿ√x)
3
The number that gives x when raised to 4
81
Value under the radical
4
Which root was taken
81
Raising the answer returns x
Results follow real-number nth-root rules. The index must be a positive integer, and even indices require x ≥ 0.
Statistics
Live readouts from your current inputs — the ladder, bars and cards all move together.
Answer ⁿ√x
3
n = 4
Radicand x
81
Under the radical
Check rootⁿ
81
Must equal x
Parity
Even index
x ≥ 0 required
Visualizations
x vs ⁿ√x
Root shrinks as index grows (x = current)
Magnitude
Formula
The nth root inverts the nth power. The index selects which root: 2 = square, 3 = cube, 4 = fourth.
Alternative Form
Variables
| Symbol | Name | Unit | Description |
|---|---|---|---|
| x | Radicand | number | Value under the radical |
| n | Index | integer | Positive integer root degree |
| ⁿ√x | Root | number | Value with (ⁿ√x)ⁿ = x |
Explanation
Higher indices compress harder: ∜81 = 3 while √81 = 9. Even indices wall off negatives; odd pass through.
Derivation
ⁿ√x = x^(1/n) since (x^(1/n))ⁿ = x. For odd n, (−y)ⁿ = −yⁿ gives ⁿ√(−x) = −ⁿ√x.
Step-by-Step Calculation
Follow each step to understand how the result is calculated from your inputs.
Read the radicand and index.
Validate the domain.
Compute as a fractional power.
Verify by raising to the index.
Confirm the check matches x.
State the final equality.
Worked Examples
See how the calculator works with different scenarios.
Scenario
Compute ∜81.
Inputs
Calculation
∜81 = 3 since 3⁴ = 81
Result
3
Interpretation
Four 3s multiply to 81.
Understanding Your Result
Your Result: Nth root (ⁿ√x): 3
Typical range: ∜81=3, ∛(−27)=−3, ⁵√32=2; higher n pushes answers toward 1.
What This Means
The result is the number that, multiplied by itself n times, returns x.
Why It Matters
Nth roots size hypercubes, invert power laws, annualize multi-year growth, and solve xⁿ = c.
Professional Interpretation
Match parity to domain: even n needs x ≥ 0; odd n takes any real x.
Recommendations
- •Verify with rootⁿ = x
- •Keep n a positive integer
- •Check parity before rooting negatives
- •Read the ladder chart to feel index growth
Learn — The Complete Guide
The nth root undoes the nth power for a chosen index. This calculator computes ⁿ√x with domain guards.
The core concept
ⁿ√x = x^(1/n): the value whose nth power is x. Parity of n decides whether negatives are allowed.
- ⁿ√x = x^(1/n): the index becomes a denominator
- Even indices need x ≥ 0; odd indices take any x
- Raising the answer to n (rootⁿ = x) proves it
Derivations — why the formulas work
Root as fractional power
- 1Let y = ⁿ√x, so yⁿ = x
- 2Raise to 1/n: y = x^(1/n) ✓
Nested fourth root
- 1∜x = x^(1/4) = (x^(1/2))^(1/2)
- 2= √(√x) ✓
Where you meet this in real life
Learn this first — prerequisites in order
- 1Square roots (n = 2) fluently
- 2Exponent law (x^a)^b = x^(ab)
Careers that use this daily
- •Financial analyst — CAGR and annualization
- •Data scientist — geometric means
- •Actuary — multi-year growth rates
- •Physicist — scaling exponents
Why it helps
- •Any index with verification
- •Explicit even/odd domain guard
- •Index ladder visualizes compression
Best practices
- •Check answers by raising to n
- •Confirm n is a positive integer
- •Use odd n for negative radicands
Limitations — what this tool does not do
- •Integer indices only (n = 1, 2, 3 …)
- •Real numbers only — no complex roots
- •Display rounds to 12 significant digits
Curriculum map — where this topic lives in school
| Level | Grades | Standards | Skills |
|---|---|---|---|
| Algebra I | 8–9 | HSN-RN | Index notation · Parity domains · xⁿ = c equations |
| Algebra II | 9–11 | HSN-RN | Fractional powers · Root estimation · Power checks |
| Precalculus / Finance | 11–College | HSF-LE | CAGR via roots · Geometric mean · Power-law inversion |
Timeline — how this idea grew up
c. 250 CE
Higher roots appear
Indian mathematicians handle cube and higher roots.
1748
Euler’s fractional powers
ⁿ√x = x^(1/n) unifies roots and exponents.
Modern
Roots run finance
CAGR and geometric means live in every spreadsheet.
A little history
- • Higher roots entered algebra through cubic equations in the 1500s.
- • Annualized growth made nth roots a finance staple.
Exam-day checklist
- • Confirm n is a positive integer first.
- • Rewrite ⁿ√x as x^(1/n) when stuck.
- • Verify: rootⁿ must equal x.
Myths vs facts
Practical Tips
- •n must be a positive integer
- •Even n needs x ≥ 0
- •Raise the answer to n to verify
- •Odd n accepts any real 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 an nth root?
How do you verify ∜81 = 3?
Why do even roots reject negatives?
Must n be an integer?
How do you estimate ∜100 mentally?
How do I use this nth root calculator?
Related Calculators
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Related Guides
Deepen your understanding with these educational resources.
Glossary
Technical terms used in mortgage calculations explained in plain language.
- Index
- The n in ⁿ√x — which root is taken.
- Radicand
- The x under the radical.
- Even root
- Root with even index — needs x ≥ 0.
- Odd root
- Root with odd index — accepts any real 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.
∜81 = 3 is a power tower
81 = 3⁴ — fourth roots peel two square roots at once: ∜81 = √(√81) = √9 = 3.
Annualized returns are nth roots
Tripling over 4 years means 3^(1/4) − 1 ≈ 31.6% yearly — every CAGR is an nth root.
Fifth roots guard codes weakly
Extracting ∛ or higher roots mod N without the key is computationally brutal.
Even roots pair up
x⁴ = 81 has solutions ±3 plus two complex ones — but ∜81 denotes only +3.
Compare Side by Side
Same radicand, rising index — each row compresses harder than the last.
| Index | x = 81 | Name | Value |
|---|---|---|---|
| n = 2 (square) | √x | 9 | |
| n = 3 (cube) | ∛x | 4.326748710922 | |
| n = 4 (fourth) | ∜x | 3 |
Recent Calculations
Your last runs on this page — stored only in this browser, never uploaded.
ⁿ√81 (n=4) = 3
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
- ▸Exponent laws (x^a)^b
Mastery progress — 0%
- I can root perfect powers for n = 2…5
- I can apply the even/odd domain rule
- I can convert ⁿ√x to x^(1/n)
- I can verify with rootⁿ = x
- 1
Week 1 · ~15 min/day
Roots by index
Goals
- Evaluate ∜81, ⁵√32 perfect powers
- State the parity domain rule
Practice
Drill 8 perfect-power roots across n = 2…5 daily.
- 2
Week 2 · ~20 min/day
Estimation and parity
Goals
- Bracket non-perfect roots
- Convert ⁿ√x to x^(1/n)
Practice
Estimate 5 non-perfect roots, then confirm here.
- 3
Week 3 · ~20 min/day
Roots in formulas
Goals
- Annualize growth with 1/n powers
- Solve xⁿ = c equations
Practice
Annualize one 3-year investment return end-to-end.
Exam tips
- •Validate n first: positive integer only.
- •Raise the answer 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.
Compute ∜16.
Compute ⁵√(−32).
Is ∜(−81) real?
Estimate ∜100.
Sales triple in 4 years. What is the annual growth factor?
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√x = x^(1/n) | Root as fractional power | ∜81 = 81^0.25 = 3 |
| (ⁿ√x)ⁿ = x | Verification identity | 3⁴ = 81 |
| Even n ⟹ x ≥ 0 | Domain guard | ∜(−16) undefined |
| Odd n ⟹ any x | Negatives allowed | ⁵√(−32) = −2 |
| ∜x = √(√x) | Fourth root nests squares | ∜81 = √9 = 3 |
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Page Information
2026-09-29
3.1.0-math
Free to use for educational and professional purposes.