FOIL With Radicals Calculator: Multiply Square Roots Step by Step
This FOIL with radicals calculator multiplies binomials that contain square roots and shows every step. Enter (2 + √3)(4 − √3), and it returns 5 + 2√3.
Use the FOIL With Radicals Calculator
Type two binomials with square roots and select Calculate. You'll see the four FOIL products, each radical simplified, like radicals combined, and the final answer.
Input tips
- Type a square root as r3, √3, or sqrt(3).
- Put the coefficient in front: 2r3 or 2*sqrt(3) means 2√3.
- Use whole numbers under the root. Variables such as √x are not supported.
- Both terms can be radicals: (√2 + √3)(√2 - √3).
- Select Clear to start a new problem.
How to FOIL With Radicals
To FOIL with radicals, multiply the binomials the usual way: First, Outer, Inner, Last. Then simplify each product and combine like radicals, which are terms with the same number under the root.
- Same root: √a · √a = a, so √3 · √3 = 3.
- Different roots: √a · √b = √(ab), so √2 · √3 = √6.
- Simplify: pull out perfect squares, so √12 = √(4 · 3) = 2√3.
- Combine only like radicals: 4√3 − 2√3 = 2√3, but √2 + √3 stays as it is.
How the FOIL With Radicals Calculator Works
Here are the steps for (2 + √3)(4 − √3):
- First: 2 · 4 = 8
- Outer: 2 · (−√3) = −2√3
- Inner: √3 · 4 = 4√3
- Last: √3 · (−√3) = −3
- Combine: 8 − 3 = 5 and −2√3 + 4√3 = 2√3
- Answer: 5 + 2√3
Conjugates: When the Radicals Disappear
The conjugate of a + √b is a − √b. Multiplying a binomial by its conjugate removes the radical, because the outer and inner products cancel:
For example, (√5 + 2)(√5 − 2) = 5 − 4 = 1. This is the same pattern as the difference of squares. It is also the key step in rationalizing a denominator: multiply the top and bottom of a fraction by the conjugate of the denominator. You can use this calculator to find the new denominator.
FOIL With Radicals Examples
| Problem | Answer | Note |
|---|---|---|
| (5 + √2)(3 + √2) | 17 + 8√2 | 15 + 2 = 17 and 5√2 + 3√2 = 8√2 |
| (3 + √5)(2 − √5) | 1 − √5 | 6 − 5 = 1 and −3√5 + 2√5 = −√5 |
| (1 + √2)(3 + 2√2) | 7 + 5√2 | 2√2 · √2 = 4 |
| (√2 + √3)(√2 − √3) | −1 | Conjugates: 2 − 3 |
| (2√3 + 1)(√12 − 1) | 11 | √12 = 2√3, so these are conjugates |
Watch for two common mistakes: adding unlike radicals (√2 + √3 is not √5) and leaving √3 · √3 as √9 instead of 3. See more common FOIL mistakes.
Try the FOIL With Radicals Calculator
FOIL first, simplify each radical, then combine like radicals. Try the FOIL with radicals calculator above with a homework problem, or practice binomials without roots in the FOIL calculator.
Häufig gestellte Fragen
Personalized questions and guidance about this specific tool.
Tool
Choose from our specialized tools to expand, factor, and master algebraic equations step-by-step.