Input coefficients and radicands
Use coefficients to scale or subtract terms. For example, enter coefficient -2 and radicand 3 for the term -2√3.
Summary of results
Enter at least one non-negative radicand and press calculate.
Use coefficients to scale, or negative coefficients to subtract terms.
Example square root addition data
These examples illustrate how coefficients and simplified radicals interact.
| Coefficients and radicands | Exact expression | Decimal sum (6 dp) |
|---|---|---|
| (1, 4), (1, 9) | √4 + √9 = 2 + 3 | 5.000000 |
| (1, 2), (2, 8) | √2 + 2√8 = √2 + 4√2 = 5√2 | 7.071068 |
| (3, 3), (-1, 12) | 3√3 - √12 = 3√3 - 2√3 = √3 | 1.732051 |
| (-2, 5), (1, 20) | -2√5 + √20 = -2√5 + 2√5 = 0 | 0.000000 |
Formula used for square root addition
The calculator handles sums of the form S = c₁√a₁ + c₂√a₂ + … + cₙ√aₙ, where each aᵢ ≥ 0 and cᵢ can be positive or negative.
For each term, the decimal contribution is cᵢ√aᵢ = cᵢ · aᵢ1/2. Integer radicands are simplified using aᵢ = k²·m so that √aᵢ = k√m and hence cᵢ√aᵢ = (cᵢk)√m.
Terms that share the same simplified radicand m are combined. For example, √8 + √2 = 2√2 + √2 = 3√2 and 3√3 - √12 = 3√3 - 2√3 = √3.
How to use this calculator
- For each row, enter a coefficient and a non-negative radicand.
- Use negative coefficients to represent subtraction between square root terms.
- Adjust the decimal places to control rounding precision.
- Press the “Calculate Sum of Square Roots” button.
- Review the exact radical expression and decimal total in the summary panel.
- Study the detailed table to see each term’s contribution and running sum.
- Use the graph and export buttons to visualise and save your calculations.
Typical inputs for square root addition
Most users experiment with small integers like 2, 3, 5, 8 and 12 to see how radicals simplify and combine into single, compact expressions.
You can also mix simple integers with non-integers, such as 2.5 or 7.3, to compare exact radical simplification against purely decimal square root behavior.
Educational uses of this calculator
The tool supports algebra lessons on radicals, making it easy to show how square roots simplify, combine, and cancel when coefficients and terms interact.
Teachers can generate instant examples, export tables as CSV files, and create handouts that highlight step-by-step running sums of square roots.
Comparison of exact and decimal results
The summary panel always displays an exact radical expression alongside a rounded decimal sum, allowing you to check how rounding affects numerical accuracy.
Higher decimal precision reveals more digits but may hide elegant simplifications, so students should compare both views when analysing answers.
Practical tips for cleaner radical expressions
When possible, choose radicands that share square factors, like 2 and 8 or 3 and 27, so their square roots simplify and combine into fewer radical terms.
If the exact expression looks complicated, try regrouping terms, checking for hidden perfect squares, or using smaller integer radicands where feasible.
Frequently asked questions
What is this square root addition calculator used for?
This tool adds terms of the form c√a. It shows both a simplified radical expression and a rounded decimal total, so you can check algebraic manipulation and numerical approximations side by side.
Can the calculator handle negative coefficients?
Yes. Enter negative coefficients to represent subtraction, such as -2 with radicand 3 for the term -2√3. The calculator includes each signed contribution in the running sum and final expression automatically.
Why do some radical expressions simplify automatically?
When the radicand contains a perfect square factor, for example 12 = 4·3, the square root splits as √12 = 2√3. The calculator automatically factors out such squares to simplify radicals.
What decimal precision should I choose for results?
For classroom demonstrations, two to four decimal places are often enough. For more sensitive numeric checks, you can increase precision up to ten places, keeping in mind that rounding may still hide exact simplifications.
Can I use non-integer radicands in my expressions?
Yes. Non-integer radicands like 2.5 or 7.3 are allowed. They are not simplified into radicals, but their square roots are still included in the decimal table, graph, and exported CSV or PDF files.
How can I export or save my square root calculations?
After calculating, use the CSV or PDF buttons above the results table. They capture every term, coefficient, radicand, decimal value, and running sum, so you can archive solutions or share worked examples quickly.