Multiplying and Simplifying Radical Expressions Calculator

Calculator

Radical index

Term A coefficient

Term A radicand

Term B coefficient

Term B radicand

Decimal precision

Multiply and Simplify Load Example Reset

Example data table

Formula used

For matching indexes, multiply outside coefficients directly. Multiply radicands inside one shared radical. Then simplify by extracting perfect powers that match the radical index.

Core rule: a√[n](b) × c√[n](d) = ac√[n](bd)

Simplification rule: if bd = qⁿr, then √[n](bd) = q√[n](r)

This calculator factors the product radicand, groups repeated prime factors by the chosen index, moves complete groups outside, and leaves leftovers inside.

How to use this calculator

1. Enter one common radical index.
2. Type the outside coefficient for each radical term.
3. Enter the radicand for each term.
4. Choose the number of decimal places.
5. Press Multiply and Simplify.
6. Review the exact result, decimal value, factors, and graph.
7. Use the export buttons for CSV or PDF copies.

Helpful notes

This page works in real-number mode. Even indexes require nonnegative radicands. Odd indexes can accept negative radicands and keep the correct sign outside the radical.

The calculator uses one shared index for both radicals. That mirrors the most common school and exam problems for multiplying and simplifying radical expressions.

FAQs

1. What does this calculator simplify?

It multiplies two radical terms with one shared index, combines coefficients, multiplies radicands, and extracts perfect powers to produce a simpler exact answer.

2. Why must the indexes match?

Direct multiplication inside one radical requires a common index. When indexes differ, you usually rewrite expressions first. This tool focuses on the standard same-index method.

3. Can I use negative coefficients?

Yes. Negative coefficients are supported and the sign is carried through the multiplication and simplification steps automatically.

4. Can I use negative radicands?

Yes, but only with odd indexes in real-number mode. Even indexes with negative radicands would produce nonreal values, so the calculator blocks them.

5. Why does the result sometimes become a whole number?

If the final radicand product is a perfect nth power, everything leaves the radical. Then the simplified result is an integer outside value.

6. What does the factor table show?

It shows the prime factorization of the product radicand, which factors were extracted as full nth-power groups, and which factors remained under the radical.

7. Is the decimal result exact?

The exact answer is the simplified radical form. The decimal line is a rounded approximation based on the precision you choose.

8. When should I use the graph?

The graph is useful when you want a quick visual comparison between the original radicands, their product, the outside factor, and the remaining inside radicand.