Direct answer
An nth root reverses an nth power; simplify a radical by extracting perfect nth-power factors, preserve an exact radical when useful, and approximate only at the reporting stage.
What this calculation tells you
Radical notation represents exact values even when their decimals never terminate. Simplification exposes perfect-power factors without changing the value.
Even real roots require nonnegative radicands, while odd roots can accept negative real values.
Where it is used
Geometry
Retain exact diagonal, distance, and radius results.
Algebra
Simplify expressions and solve supported radical relationships.
Science and engineering
Work with inverse power laws and dimensional relationships.
Measurement
Separate exact model output from the precision needed for use.
When this guide helps
- Simplifying √72.
- Comparing an exact radical with a decimal.
- Checking whether a real root exists.
- Avoiding premature rounding in later calculations.
Identify perfect-power factors
Write the radicand as a product containing the largest convenient perfect power, then move its root outside. The remaining radical must preserve the original sign and index.
Check the index and sign
An even index has two equation roots for x^n=a when a is positive, but the radical symbol denotes the principal nonnegative root. Odd roots preserve the radicand's sign.
Choose exact or approximate output
Exact radicals support algebraic checking and prevent compounded rounding. A measured or manufactured result may later need a decimal at a stated resolution.
Common mistakes
Typical errors include splitting a root across addition and writing ± as part of the principal radical value.
- Factor before approximating.
- Check index parity.
- Label principal root versus equation solutions.
Worked case: simplify a square root
Simplify sqrt(72).
Factor 72=36x2, so sqrt(72)=sqrt(36)sqrt(2)=6sqrt(2).
The exact result is 6sqrt(2), approximately 8.485.
The exact radical and decimal serve different purposes; rounding should not replace the exact form prematurely.
Reproduce this worked caseOpen Radical Simplifier Calculator
Worked case: odd root of a negative number
Find the real cube root of -125.
Because (-5)³=-125, cube root(-125)=-5.
The real result is -5.
Odd roots accept negative real radicands, while an even root of a negative number is not real.
Reproduce this worked caseOpen Radical Simplifier Calculator
Compare roots and radicals cases before generalising
State the number system and principal-root convention. Squaring sqrt(x) returns x on its domain, but sqrt(x²)=|x| for real x, not always x.
| Radical | Exact form | Approximation/domain |
|---|---|---|
| sqrt(72) | 6sqrt(2) | ≈8.485 |
| cube root(-125) | -5 | Real |
| sqrt(-1) | Not real | Complex i if allowed |
roots and radicals: calculation checklist
- Factor perfect powers
- Check even-root domain
- Keep principal-root convention
- Retain exact form
- Verify by powering
Practical questions
Frequently asked questions
Is √9 equal to ±3?
The principal radical √9 equals 3. The equation x²=9 has two solutions, x=±3.
Can radicals be exact?
Yes. √2 is an exact value even though its decimal expansion is nonterminating.
Why can an odd root accept a negative number?
An odd power preserves sign, so a negative real number has one negative real odd root.
Further reading
Authoritative sources
Use these primary and professional resources to check definitions, conventions, or requirements that may extend beyond this guide.