Input
Each binomial has two terms: c√r + d√s. Use negative coefficients for minus. Set radicand 1 to enter a rational term.
Binomial A
Binomial B
Result
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Steps
History
| # | Binomial A | Operation | Binomial B | Result | ≈ Decimal |
|---|
Formulas Used
Add/Subtract: Like radicals combine: (a√m + b√n) ± (c√m + d√n) = (a±c)√m + (b±d)√n, after simplifying each radical.
Multiply (FOIL): (a√m + b√n)(c√p + d√q) = ac√(mp) + ad√(mq) + bc√(np) + bd√(nq), then simplify radicals and combine like terms.
Divide (Conjugate): \nA÷B = A·conj(B) / (x² − y²), where B = x + y with x = b₁√d₁ and y = b₂√d₂. Denominator becomes b₁²d₁ − b₂²d₂ (rational).
Simplify √n: Extract largest square factor s²|n ⇒ √n = s√(n/s²). Combine terms having identical simplified radicands.
How To Use
- Enter coefficients and radicands for both binomials. Use negatives as needed.
- Choose an operation: add, subtract, multiply, or divide with rationalization.
- Click Calculate. Review the simplified expression, steps, and decimal approximation.
- Click Download CSV or Download PDF to save your results.
Example Data
| # | Binomial A | Op | Binomial B | |
|---|---|---|---|---|
| 1 | (3√8 + 2√2) | + | (1√2 + 5√8) | |
| 2 | (4√18 − 1√8) | × | (2√2 + 1√18) | |
| 3 | (5√5 + 3√20) | ÷ | (2√5 − 1√20) |
FAQs
Two radical terms are “like” if their simplified radicands are identical. Only like radicals can be combined through addition or subtraction.
Simplifying exposes like radicals by extracting square factors, enabling correct combination and cleaner results.
We apply FOIL, simplify each radical product, then combine terms that share the same simplified radicand.
We multiply numerator and denominator by the conjugate of the denominator. The denominator becomes rational by the difference of squares identity.
Yes. They’re converted to exact fractions internally to avoid rounding. Results display as simplified fractions when appropriate.
Any term with √0 equals zero. It contributes nothing to the final expression.
Common Radical Simplifications (√n → a√b)
Quick reference for frequently used square roots after extracting the largest square factor.
| n | Largest square factor | Simplified form | √n (decimal) |
|---|
Operation Data Examples (Auto‑computed)
Representative inputs and the simplified outcomes using the same engine as the calculator.
| # | Binomial A | Op | Binomial B | Simplified Result | ≈ Decimal |
|---|