Calculator
Use integer entries for exact simplification. Leave unused factors fully blank.
Example Data Table
| Expression | Common Index | Simplified Product | Decimal Approximation |
|---|---|---|---|
| √2 × √8 | 2 | 4 | 4 |
| 2√3 × 3√12 | 2 | 36 | 36 |
| 2∛4 × 3∛2 | 3 | 12 | 12 |
| (3√5)/2 × (4√20)/3 | 2 | 20 | 20 |
| √3 × ∛9 | 6 | 3×6√3 | 3.602810865 |
Formula Used
For each factor, the calculator reads a coefficient, a root index, a radicand, and an outside denominator. A general factor is written as:
(c × n√a) / d
To multiply several roots, the calculator first multiplies all outside coefficients together. It also multiplies all denominators together. After that, it finds the least common multiple of all root indices. This creates one shared index that can be used for every radical term.
If the common index is L, then each radicand is rewritten as a^(L/n). The calculator multiplies those rewritten radicands into one combined value under a single radical. The working structure becomes:
(product of coefficients / product of denominators) × L√(product of rewritten radicands)
Next, the calculator simplifies the combined radical. It factorizes the combined radicand and pulls out every perfect Lth power. Those extracted values move outside the radical. The remaining value stays inside the radical. The final exact form is then reduced with any common factor between numerator and denominator.
The decimal result is calculated by evaluating every root numerically and multiplying the factor values in order.
How to Use This Calculator
Start by filling in one or more factor boxes. Enter the outside coefficient first. If there is no outside coefficient, keep it as 1. Enter the root index next. Use 2 for square roots, 3 for cube roots, and higher values for advanced roots.
Then enter the radicand for each factor. The denominator field is for any outside fraction. If your factor is not divided by anything, keep the denominator as 1. You can leave unused factors blank. The calculator only multiplies the factors you actually fill in.
After you press Multiply Roots, the result appears above the form. You will see the common index, combined radicand, common radical form, exact simplified answer, decimal answer, and a clear sequence of working steps. This is useful for checking homework, teaching radical rules, or preparing worked examples.
The export buttons create a CSV summary and a PDF report of the current result. The graph also helps you compare each factor’s numeric value with the cumulative product as the multiplication develops from one factor to the next.
About Multiplying Roots
Why root multiplication matters
Multiplying roots is a common skill in algebra, precalculus, and technical problem solving. Students see it when simplifying radical expressions, solving geometry formulas, and working through equations that contain irrational terms. Engineers, analysts, and teachers also use radical multiplication while converting formulas into exact symbolic forms before switching to decimal evaluation. A reliable calculator can shorten that process while still showing the reasoning behind the answer.
How exact answers are built
Many learners only multiply square roots that already share the same index. That is the easiest case because the radicands can be multiplied directly under one radical. More advanced examples mix square roots, cube roots, and higher-order roots. In that situation, a shared index is needed. The calculator handles that step automatically by using the least common multiple of the indices. Once all roots are rewritten to the same index, the radicands can be merged into one product and simplified.
Why simplification is useful
Simplifying the final radical makes the expression easier to read and compare. For example, a large radicand may contain a perfect square, perfect cube, or another perfect power that can move outside the radical. Pulling those factors out often reveals a cleaner exact answer and helps students recognize structure in the problem. This matters in exam work because exact forms are usually preferred before rounding.
Exact form and decimal form together
This calculator shows both forms because they serve different purposes. The exact result preserves algebraic accuracy. The decimal result is helpful for checking magnitude, estimating comparisons, and verifying manual work. By viewing the step list, the example table, and the graph together, users can understand not only the final product but also the transformation that produced it.
FAQs
1. What does it mean to multiply roots?
It means multiplying radical expressions such as √2 × √8 or 2∛4 × 3∛2. The calculator combines outside numbers, rewrites indices when needed, and simplifies the final radical.
2. Can this calculator handle different root indices?
Yes. It finds the least common multiple of the indices, rewrites each radical to that common index, and then multiplies the radicands in one exact structure.
3. Why should I use integer inputs?
Integer values keep the exact symbolic simplification dependable. They let the calculator factor the combined radicand and extract perfect powers cleanly from the radical.
4. What happens if one factor equals zero?
If a coefficient is zero, the whole product becomes zero. If a radicand is zero, that radical factor is zero too, so the total product is also zero.
5. Does it simplify only square roots?
No. It also simplifies cube roots and higher-order roots. The exact same method works after the calculator converts all radicals to a shared index.
6. Why is the decimal answer included?
The decimal answer helps you compare sizes, check manual calculations, and estimate values quickly. It is useful when the exact radical still contains an irrational part.
7. What is the common index?
The common index is the least common multiple of all entered root indices. It allows unlike radicals to be rewritten in one compatible multiplication form.
8. Can I export the current result?
Yes. After calculation, you can download a CSV summary or create a PDF report that includes the expression, exact result, decimal value, and worked steps.