Input
Result
Steps
Click “Compute” to see detailed steps.
Options
| # | Base | m | n | Outside factor | Radical index | Radicand | Final expression | Format | Assumptions |
|---|
Batch Mode
Paste one case per line as base,m,n (e.g., 18 x^5 y^3,3,2). Current options apply to all rows.
Example Data
| Base | m | n | Radical Form (simplified) |
|---|---|---|---|
| 18 x^5 y^3 | 3 | 2 | 27 x^7 y^4 √(2 x y) |
| 81 x^4 | 1 | 2 | 9 x^2 |
| -8 | 1 | 3 | -∛(8) = -2 |
Formula Used
For a rational exponent m/n with integers m and n>0, am/n = √[n]{am}. Splitting integer and remainder via the Euclidean division m = q n + r with 0 ≤ r < n gives am/n = aq · √[n]{ar}.
For a monomial A · ∏ viki, raising to m/n multiplies exponents by m and the extraction uses q = ⌊(ki·m)/n⌋ and r = (ki·m) mod n per variable. Numeric factors use prime factorization to pull out perfect n-th powers.
Real-only mode is assumed. If variables may be negative and n is even, absolute values can be required unless you assume nonnegativity.
How to Use
- Enter the base expression as integers and variables like x^k.
- Set the numerator m and denominator n of the rational exponent.
- Pick options: display format, variable assumptions, denominator rationalization, approximation.
- Click Compute to see the simplified radical form and steps.
- Use Add to Table or run Batch Mode for multiple rows.
- Export your table using Download CSV, Download PDF, or Download JSON.
FAQs
Absolute Values in Radical Form (Even vs Odd Index)
Absolute values appear when taking an even-index root of a power in variables whose sign is unknown.
- √(x^2) = |x|, and in general √[n]{x^n} = |x| when n is even.
- If n is odd, √[n]{x^n} = x (no absolute value needed).
- Extraction example: √(x^3) = √(x^2·x) = |x| √(x). With “Assume nonnegative” enabled, this simplifies to x √(x).
- For multi-variable monomials, each variable follows the same rule independently.
| Expression | Index | Result (unknown sign) | With “Assume nonnegative” |
|---|---|---|---|
| √(x^4) | 2 | |x|^2 | x^2 |
| √[4]{y^8} | 4 | |y|^2 | y^2 |
| √(x^3) | 2 | |x| √(x) | x √(x) |
Tip: Toggle the “Assume nonnegative” option in this tool to include or drop absolute values when n is even.
Coefficient Extraction by Prime Factorization — Quick Examples
To pull perfect n-th powers outside the radical, factor the numeric coefficient into primes. For each prime p with exponent e:
Outside exponent: ⌊e/n⌋ Inside exponent: e mod n
| Number | n | Prime factorization | Outside | Inside | Result |
|---|---|---|---|---|---|
| 72 | 2 | 2^3 · 3^2 | 2 · 3 = 6 | 2 | 6 √(2) |
| 45 | 2 | 3^2 · 5 | 3 | 5 | 3 √(5) |
| 216 | 3 | 2^3 · 3^3 | 2 · 3 = 6 | 1 | 6 |
| 80 | 4 | 2^4 · 5 | 2 | 5 | 2 √[4]{5} |
The same rule applies when coefficients multiply variables—extract numeric and variable parts independently, then recombine.