What Is Binomial Radical Expressions?
To multiply radical binomials, use FOIL. Multiply the numbers outside the roots together and the numbers under the roots together, then simplify each radical and add the like ones.
For example, (2√3 + 5)(√3 − 4) gives 6 − 8√3 + 5√3 − 20, which simplifies to −14 − 3√3. Only radicals with the same number under the root can be added.
Type each expression with √ or sqrt and whole numbers. The calculator also gives a decimal check so you can confirm the exact answer.
How to Use the Binomial Radical Expressions Calculator
- Type each expression, for example 2√3 + 5.
- Read each partial product and its simplified form.
- Check the combined answer and its decimal value.
Binomial Radical Expressions Formula
Product of roots
Conjugates
Binomial Radical Expressions Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| a√b | A term with a number a outside the root and b inside | |
| Radicand | The number under the root sign | |
| Like radicals | Radicals with the same radicand | Can be added |
| Conjugate | The same binomial with the middle sign flipped | Product has no radical |
Binomial Radical Expressions Worked Example
With the example values loaded in the calculator above, the steps are:
- First: 2√3 + 5. Second: √3 − 4.
- Multiply each term of the first by each term of the second. Multiply the numbers outside the roots together, and the numbers under the roots together.
- (2√3)(√3) = 2√9 = 6
- (2√3)(−4) = -8√3 = −8√3
- (5)(√3) = 5√3 = 5√3
- (5)(−4) = -20 = −20
- Simplify each radical by pulling out perfect squares, then add like radicals.
- Result: −14 − 3√3.
Result: Product (simplified) = −14 − 3√3.
Worked example: (2√3 + 5)(√3 − 4)
First: 2√3 · √3 = 2 · 3 = 6. Outer: 2√3 · (−4) = −8√3. Inner: 5 · √3 = 5√3. Last: 5 · (−4) = −20.
Combine: the whole numbers give 6 − 20 = −14, and the radicals give −8√3 + 5√3 = −3√3. The product is −14 − 3√3, about −19.1962.
Conjugates remove the radical
A binomial times its conjugate, the same terms with the sign of one term flipped, has no radical left, since the middle terms cancel. This is how you rationalize denominators.
Where Binomial Radical Expressions Is Used
Algebra homework
Multiply and simplify radical binomials with steps.
Rationalizing denominators
Practise multiplying by a conjugate.
Checking answers
Confirm an exact radical answer with a decimal check.
Common Binomial Radical Expressions Mistakes to Avoid
- Pull out perfect squares: √12 = 2√3.
- Only add radicals with the same radicand.
Binomial Radical Expressions Terms and Definitions
- Radicand
- The number under the root sign.
- Conjugate
- The same binomial with the sign between the terms changed.
Binomial Radical Expressions FAQs
How do you multiply radical binomials?
Use FOIL. Multiply outside numbers together and radicands together, then simplify.
When can I add radicals?
When they have the same number under the root, like 3√5 and 2√5.
What is √a times √a?
It is a, because the root and the square cancel.
What happens with conjugates?
The cross terms cancel and the radicals vanish, leaving a² − b.