Multiplying Radical Expressions Calculator

Expression A	Expression B	Exact Product	Decimal Approximation
2√(3)	4√(12)	48	48
3√(2)	5√(8)	60	60
2∛(4)	3∛(2)	12	12
4√(5)	2√(15)	40√(3)	69.2820323
6√(18)	2√(2)	72	72

Formula Used

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

Simplification rule: √[n](xⁿy) = x√[n](y)

Different indices: convert both radicals to a common index using the least common multiple, then multiply and simplify.

This page also shows a decimal approximation using standard root values for quick comparison.

Frequently Asked Questions

1. Does this calculator simplify radicals automatically?

Yes. When indices match, the calculator multiplies radicands and simplifies extracted perfect powers. When indices differ, it converts them to a common index when practical and still returns a decimal product.

2. Can I use cube roots or higher roots?

Yes. It works with square roots, cube roots, and higher indices. Enter the correct index for each radical, and the calculator handles the structure automatically.

3. Are negative values allowed?

Negative coefficients are allowed because they sit outside the radical. Radicands should remain nonnegative here, which keeps the exact simplification clear and avoids complex-number cases.

4. What is the difference between exact and decimal results?

The exact result keeps the radical form whenever possible. The decimal result shows an approximation, which is useful for checking size, comparison, and graphing.

5. Why do some products become whole numbers?

When an extracted factor forms a perfect power, it moves outside the radical. That is why products like √8 × √2 become 4 instead of √16.

6. Will different indices always simplify fully?

Usually, yes. If the transformed radicand becomes extremely large, the calculator may stop full symbolic simplification and still provide the correct decimal product and conversion steps.

7. What does the graph show?

The graph compares the decimal values of expression A, expression B, and the final product. It helps you see scale changes caused by multiplication.

8. What inputs work best?

Use integers for clean symbolic simplification. Very large values can create huge transformed radicands, especially with different indices, so moderate inputs work best for readable exact output.