Multiply Radicals Calculator
Multiply radical expressions with automatic simplification and step-by-step solutions
Common Examples
√2 × √8 = 4
3√5 × 2√5 = 30
2√3 × 4√6 = 24√2
√12 × √18 = 6√6
Multiplying Radicals
When multiplying radicals, you multiply the coefficients together and the radicands together separately. The key property is: √a × √b = √(ab), which applies to all radicals with the same index.
Steps to Multiply Radicals
- Multiply the coefficients (numbers outside the radical)
- Multiply the radicands (numbers inside the radical)
- Combine: (coefficient) × radical(radicand)
- Simplify the resulting radical if possible
Product Property of Radicals
For radicals with the same index n:
ⁿ√a × ⁿ√b = ⁿ√(a × b)
Frequently Asked Questions
How do you multiply two radicals?
To multiply radicals with the same index, multiply the coefficients together and multiply the radicands together. For example: (3√2)(4√3) = 12√6. Always simplify the result if possible.
Can you multiply radicals with different indices?
No, you cannot directly multiply radicals with different indices using the simple product rule. You would need to convert them to exponential form first. For example, √2 and ∛3 have different indices.
What is the product property of radicals?
The product property states that √a × √b = √(ab). This means you can multiply the numbers under the radicals together. This property works for all radicals with the same index.
Do you simplify before or after multiplying?
You can do either, but it's often easier to multiply first and then simplify the result. However, simplifying first can sometimes make the multiplication easier if you recognize common factors.
How do you multiply a radical by itself?
When you multiply a radical by itself, you're squaring it. √a × √a = (√a)² = a. The radical disappears and you're left with the radicand. For example: √5 × √5 = 5.
What about multiplying binomials with radicals?
Use the FOIL method (First, Outer, Inner, Last) to multiply binomials containing radicals. For example: (√2 + √3)(√2 + √5) = 2 + √10 + √6 + √15. Combine like terms if possible.
Can the result be a whole number?
Yes! When the product of the radicands is a perfect square (or perfect nth power), the result simplifies to a whole number. For example: √2 × √8 = √16 = 4.
What if one coefficient is negative?
Follow normal multiplication rules for signs. A positive times a negative gives a negative result. For example: (−2√3)(5√2) = −10√6.
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