Enter up to six radical terms
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Example data table
| Expression | Simplified Terms | Final Result | Why It Works |
|---|---|---|---|
| 3√8 - 2√18 + 5√2 | 6√2 - 6√2 + 5√2 | 5√2 | Each term simplifies to the same radical, so coefficients combine directly. |
| 4∛54 + 2∛16 | 12∛2 + 4∛2 | 16∛2 | Both cube roots reduce to the same inside radicand after extracting perfect cubes. |
| 5√12 - 3√3 | 10√3 - 3√3 | 7√3 | √12 simplifies to 2√3, making the radicals like terms. |
Formula used
1. Simplify each radical first by extracting perfect powers:
a × n√(mnb) = am × n√b
2. Combine only like radicals:
a√b ± c√b = (a ± c)√b
3. For higher roots, the index must also match:
a × n√b ± c × n√b = (a ± c) × n√b
If two terms do not simplify to the same index and radicand, they remain separate in the result.
How to use this calculator
- Enter each term’s coefficient, root index, and radicand.
- Choose plus or minus for every term after the first.
- Click Calculate Expression to simplify and combine.
- Review the result, steps, table, and graph above the form.
- Download the summary as CSV or PDF when needed.
FAQs
1. What are like radicals?
Like radicals have the same root index and the same simplified radicand. Only those terms can be combined by adding or subtracting their coefficients.
2. Why must I simplify radicals first?
A radical may hide a perfect square, cube, or higher power. Simplifying first can reveal matching radicals that were not obvious in the original expression.
3. Can unlike radicals be added directly?
No. If two terms have different simplified radicands or different indices, they stay separate. The expression may still be simplified term by term, but not combined fully.
4. Does this calculator work for cube roots and higher roots?
Yes. Enter the desired root index for each term. The calculator extracts perfect powers based on that index, then checks whether the simplified radicals are like terms.
5. What happens if the radicand is zero?
Any radical with radicand zero equals zero, regardless of coefficient. That term contributes nothing to the final expression and is shown clearly in the steps.
6. Can coefficients be decimals?
Yes. Decimal coefficients are allowed. The calculator keeps the radical simplification exact while combining coefficients numerically in the final result.
7. Why can a radical become a whole number?
If the entire radicand is a perfect power for the chosen index, the radical simplifies completely. For example, √49 becomes 7 and no radical remains.
8. How can I verify the answer manually?
Simplify each term separately, rewrite them with matching radicals where possible, then add or subtract only the coefficients of the like radicals.