Adding and Subtracting Radical Expressions

Example data table

You can import these rows via the Load Example button above.

Formulas used

Radicals are simplified by extracting perfect powers from the radicand. If \(r=\prod p_i^{e_i}\) and we take an \(n\)-th root, then

\( \sqrt[n]{r} \;=\; \Big(\prod p_i^{\lfloor e_i/n \rfloor}\Big)\,\sqrt[n]{\prod p_i^{\,e_i \bmod n}} \).

Like radicals combine only when the index and remaining radicand match:

\( a\sqrt[n]{b} \pm c\sqrt[n]{b} \;=\; (a\pm c)\sqrt[n]{b} \).

Even indices require non‑negative radicands. Negative radicands are valid for odd indices.

How to use this calculator

1. Type an expression using √r, sqrt(r), or root(n,r); then press Parse Expression.
2. Or build terms in the table: choose sign, coefficient, index, and radicand. Press Add Term.
3. Click Calculate to simplify each term, group like radicals, and combine them.
4. Read the Result as an exact symbolic sum and a decimal approximation.
5. Export your inputs and results using Download CSV or Download PDF.

Tip: For decimals inside radicals, convert to integers first, or use fractional forms.

How to simplify radical expressions with variables (add & subtract)

When variables appear under an \(n\)-th root, factor numerical and variable parts into perfect \(n\)-th powers and a remainder. For any nonnegative symbol \(a\) and integer \(k\),

\( \sqrt[n]{a^{k}} \;=\; a^{\lfloor k/n \rfloor}\,\sqrt[n]{\,a^{\,k \bmod n}\,} \quad\) (assume \(a\ge 0\) when \(n\) is even).

• Extract perfect powers: pull outside factors where exponents are multiples of \(n\); keep the remainder inside.
• Normalize radicands: sort variables alphabetically and reduce exponents modulo \(n\) so like terms match exactly.
• Combine like radicals: add/subtract only if the index and the entire remaining radicand (including variable exponents) are identical.
• Domain note: For even roots, \(\sqrt{x^2}=|x|\). If you assume variables are nonnegative, then \(\sqrt{x^2}=x\).
• Signs with odd roots: negative radicands are allowed for odd \(n\).

Worked example (square roots)

\( 3x\sqrt{12x^{3}y^{4}} \;-\; 2x\sqrt{3xy^{4}} \;=\; 3x\,(2xy^{2}\sqrt{3x}) \;-\; 2x\,(y^{2}\sqrt{3x}) \)
\( \qquad =\; (6x^{2}y^{2} - 2xy^{2})\sqrt{3x} \;=\; 2xy^{2}(3x-1)\sqrt{3x}. \)

Worked example (cube roots)

\( \sqrt[3]{54a^{5}b^{2}} \;+\; 4\sqrt[3]{2a^{2}b^{5}} \;=\; 3ab\sqrt[3]{2a^{2}b^{2}} \;+\; 4b\sqrt[3]{2a^{2}b^{2}} \)
\( \qquad =\; b(3a+4)\sqrt[3]{2a^{2}b^{2}}. \)

Checklist for adding/subtracting with variables

1. Fully simplify each radical by extracting perfect \(n\)-th powers of numbers and variables.
2. Write remaining radicands in a consistent order with reduced exponents.
3. Combine coefficients only for terms with identical index and identical remaining radicand.

This tool focuses on numeric radicands; the rules above show how to simplify and combine symbolic variable radicals by hand before combining.

When can radicals be combined?

Rule: Radical expressions can be combined (added or subtracted) when they are like radicals — that is, after full simplification they have the same index and an identical remaining radicand.

• First extract perfect \(n\)-th powers from each term (numbers and variables).
• Ensure variable exponents are reduced modulo \(n\) and ordered consistently.
• Then add/subtract the coefficients of matching terms only.
• Even indices require nonnegative radicands; otherwise use absolute values as needed.

Example: \(5\sqrt{18}-\sqrt{8}+2\sqrt{50}= (15-2+10)\sqrt{2}=23\sqrt{2}\).

Example: Using the calculator (step by step)

Expression:  \(3\sqrt{8} + 2\sqrt{18} - \sqrt{2} - 5\sqrt{50}\)

1. Paste the expression above into the input box and click Parse Expression.
2. Click Calculate. The tool extracts perfect squares and groups like radicals.
3. Read the Simplification steps and the combined Result.

Algebraic simplification shown by the tool:

\( 3\sqrt{8}=3\cdot 2\sqrt{2}=6\sqrt{2},\quad 2\sqrt{18}=2\cdot 3\sqrt{2}=6\sqrt{2},\quad \sqrt{50}=5\sqrt{2}.\)
\( 6\sqrt{2}+6\sqrt{2}-\sqrt{2}-5\sqrt{2} = (6+6-1-5)\sqrt{2} = 6\sqrt{2}. \)

Now include the last term \(-5\sqrt{50}\):

\( 6\sqrt{2} - 25\sqrt{2} = -19\sqrt{2}\).

Another example (with variables, done by hand)

\( 3x\sqrt{12x^{3}y^{4}} - 2x\sqrt{3xy^{4}} = 3x(2xy^{2}\sqrt{3x}) - 2x(y^{2}\sqrt{3x}) = 2xy^{2}(3x-1)\sqrt{3x}. \)

The calculator groups numeric radicals automatically; variable examples follow the same rules after hand-simplifying to like radicals.

Adding and Subtracting Radical Expressions — Worksheet

Assume all variables represent nonnegative real numbers for even roots so absolute values are not required after extraction.