Simplify tough radicals using clean factor extraction. See exact forms, imaginary units, decimals, and verification. Download reports and practice examples for faster algebra mastery.
The chart compares the radicand magnitude, extracted square factor, remaining radical factor, and overall magnitude of the simplified result.
| Outside Coefficient | Radicand | Exact Simplified Form | Approximate Value |
|---|---|---|---|
| 1 | -12 | 2i√3 | 3.464102i |
| 3 | -50 | 15i√2 | 21.213203i |
| -2 | 72 | -12√2 | -16.970563 |
| 1 | 49 | 7 | 7 |
| 4 | -1 | 4i | 4i |
This simplify complex square roots calculator helps students rewrite radicals into exact standard form. It handles positive radicands, negative radicands, and values that already contain an outside coefficient. When the radicand is negative, the calculator converts the expression into an imaginary-number form using i. It then removes every possible perfect-square factor from under the radical so the final answer is fully simplified.
The calculator is useful in algebra, pre-calculus, and early complex-number practice. Many learners can estimate square roots but still struggle to express them exactly. This page solves that issue by showing the prime factorization, the largest perfect-square factor, the extracted multiplier, and the final exact result. You also get a decimal approximation so exact and approximate values can be compared quickly.
Advanced support comes from the step-by-step logic. Instead of returning only an answer, the tool explains how the radical was broken apart. That makes it easier to check classwork, understand homework patterns, and confirm whether a radical is already simplified. The graph offers another view by comparing the original magnitude with the extracted square factor and remaining radical factor.
The included export options make the result portable. You can download a CSV file for records or worksheet building, and you can also download a PDF summary for printing or sharing. The example table below the calculator gives quick practice references that show how real and imaginary radicals behave under simplification.
For positive radicands: a√n = a√(s²r) = as√r
For negative radicands: a√(-n) = ai√n = ai√(s²r) = asi√r
Here, s² is the largest perfect-square factor of the absolute radicand, and r is the remaining factor left inside the radical. If r = 1, the radical disappears completely.
It means rewriting a radical with a negative radicand into standard form using i and removing every perfect-square factor from inside the root.
Because sqrt(-1) equals i. Any negative square root can be rewritten as sqrt(-1) times sqrt of the positive part, which produces an imaginary result.
If the radicand is a perfect square, the radical disappears completely. For negative perfect squares, the answer becomes a whole-number multiple of i.
Yes. It simplifies ordinary square roots by extracting the largest perfect-square factor and leaving only the non-square remainder under the radical.
The exact answer keeps the radical form, which is mathematically precise. The decimal answer is an approximation based on your selected precision setting.
The outside coefficient multiplies the entire square-root expression. After simplification, it combines with the extracted factor to build the final exact result.
The calculator factors the absolute radicand into primes, pairs matching factors, and multiplies those pairs to create the largest perfect-square factor.
A radical is fully simplified when no perfect-square factor larger than 1 remains inside the root and any negative sign has been converted into i.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.