Free Radicals Calculator with Steps
Simplify radicals, rationalize denominators, add, multiply and divide with animated factor trees, interactive number lines, and step-by-step working. Perfect for AP and SAT math exams.
Enter a positive number to simplify (e.g., 72, 50, 200)
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What is a Radical?
A radical is an irrational number expressed as a root that cannot be simplified to remove the root sign. The symbol √ is called the radical sign, and the number underneath is called the radicand.
Formally, √a is a radical when a is a positive integer that is NOT a perfect square. This means √a is irrational — its decimal expansion goes on forever without repeating.
For example, √2, √3, √5, √7 are all radicals because they cannot be written as exact fractions. However, √4 = 2, √9 = 3, and √25 = 5 are NOT radicals — they simplify to whole numbers.
In AP and high school math, you are expected to leave answers in "exact form" using radicals, rather than writing decimal approximations. Writing 6√2 is exact; writing 8.485... is not.
| Expression | Value | Radical? |
|---|---|---|
| √2 | 1.41421... | Yes |
| √4 | 2 | No |
| √7 | 2.64575... | Yes |
| √9 | 3 | No |
| 3√5 | 6.70820... | Yes |
| √16 | 4 | No |
Radical Laws & Rules
Product Rule
√(ab) = √a × √b
Split radicals by multiplying radicands
Quotient Rule
√(a/b) = √a / √b
Divide radicals by dividing radicands
Like Radicals Addition
a√n + b√n = (a+b)√n
Combine coefficients of like radicals
Squaring Rule
(√a)² = a
Squaring a radical removes the root
Conjugate Rule
(a+√b)(a−√b) = a²−b
Difference of squares eliminates radicals
Rationalization
Multiply by √n/√n or conjugate
Remove radicals from the denominator
Types of Radicals
| Type | Notation | Example | Can Combine? |
|---|---|---|---|
| Simple radical | √n | √2, √5, √13 | N/A |
| Like radicals | a√n, b√n | 2√3 and 5√3 | Yes — add coefficients |
| Unlike radicals | a√m, b√n | 2√3 and 4√5 | No — different radicands |
| Compound radical | a + b√n | 2 + 3√5 | Only the rational parts |
| Binomial radical | (a + b√n) | (1 + √2) | Expand using FOIL |
| Conjugate radicals | (a+√b) and (a−√b) | (3+√2) and (3−√2) | Multiply → a²−b |
Perfect Squares Reference
Memorizing these squares helps you quickly identify factors when simplifying radicals. If the number under the √ is divisible by any of these, you can simplify.
Which Method Do I Use?
Is it a single √n?
→ Simplify
Find the largest perfect square factor and extract it.
Same number under the √?
→ Add/Subtract coefficients
Like radicals: a√n ± b√n = (a±b)√n. Simplify first if needed.
Radical in the denominator?
→ Rationalize
Simple: multiply by √n/√n. Binomial: multiply by conjugate.
(a+b√c)(d+e√f)?
→ Expand with FOIL
First, Outside, Inside, Last. Then collect like radicals.
Common Mistakes to Avoid
Adding Unlike Radicals
√2 + √3 = √5
√2 + √3 (cannot simplify)
Only like radicals combine. √2 + √3 stays as it is.
Addition Instead of Product Rule
√12 = √4 + √8
√12 = √4 × √3 = 2√3
Use √(ab) = √a × √b (multiplication), never addition.
Forgetting to Simplify First
"√8 + √2 can't add"
√8 = 2√2, so √8 + √2 = 3√2
Always simplify radicals before trying to add or subtract.
Radical Left in Denominator
5/√2 as final answer
5/√2 = 5√2/2
Always rationalize — exams require rational denominators.
Wrong Conjugate Sign
Conjugate of (3+√2) is (3+√2)
Conjugate of (3+√2) is (3−√2)
The conjugate changes the sign between the terms.
Not Fully Simplifying
Leaving √12 as final answer
√12 = √(4×3) = 2√3
Check: does the radicand have perfect square factors?
Worked Examples
Simplify √200
Step 1: Find perfect square factors of 200
200 = 100 × 2 (100 is the largest)
Step 2: Apply √(ab) = √a × √b
√200 = √100 × √2
Step 3: Simplify √100 = 10
= 10 × √2
Answer: 10√2
Rationalize 6/√3
Step 1: Multiply top and bottom by √3
(6/√3) × (√3/√3)
Step 2: Numerator: 6 × √3 = 6√3
Step 3: Denominator: √3 × √3 = 3
= 6√3 / 3
Step 4: Simplify the fraction
Answer: 2√3
Rationalize 5/(2+√3)
Step 1: Identify conjugate of (2+√3) → (2−√3)
Step 2: Multiply top and bottom by (2−√3)
5(2−√3) / (2+√3)(2−√3)
Step 3: Expand numerator: 10 − 5√3
Step 4: Denominator: 2² − (√3)² = 4 − 3 = 1
Answer: 10 − 5√3
Expand (3+√2)(1−2√2)
Step 1: FOIL — First: 3 × 1 = 3
Step 2: Outside: 3 × (−2√2) = −6√2
Step 3: Inside: √2 × 1 = √2
Step 4: Last: √2 × (−2√2) = −2 × 2 = −4
= 3 − 6√2 + √2 − 4
Step 5: Collect like terms
Answer: −1 − 5√2
Radicals vs Decimals — Why Use Exact Form?
AP and SAT scoring guides award marks for exact (radical) form. Decimals are approximations that lose precision — radicals capture the exact value.
| Radical (Exact) | Decimal (Approximate) | Exact? |
|---|---|---|
| √2 | 1.41421356... | No — infinite decimal |
| 3√5 | 6.70820393... | No — infinite decimal |
| 6√2 | 8.48528137... | No — infinite decimal |
| 2√3 | 3.46410162... | No — infinite decimal |
| 10√2 | 14.14213562... | No — infinite decimal |
Decimal values go on forever without repeating. Only radical form is exact.
Frequently Asked Questions
What is a radical?
A radical is an irrational number expressed as a root that cannot be simplified. √2 is a radical, but √4 = 2 is not. The symbol √ is the radical sign and the number underneath is the radicand.
How do you simplify radicals?
Find the largest perfect square factor, use √(ab) = √a × √b to split it, then simplify the perfect square part. For example, √72 = √(36×2) = 6√2.
What are like radicals?
Like radicals have the same radicand (number under the root). You can only add/subtract like radicals: 2√3 + 5√3 = 7√3. Unlike radicals (e.g. √2 + √3) cannot be combined.
Why rationalize the denominator?
Rationalizing produces cleaner answers that are easier to compare and combine. On AP and SAT exams, rationalized answers are the expected standard form — leaving a radical in the denominator is incomplete.
What is a conjugate?
The conjugate of (a + √b) is (a - √b). Multiplying conjugates uses the difference of squares to eliminate radicals: (a + √b)(a - √b) = a² - b.
Is this aligned with AP and SAT math?
Yes! It covers simplifying, adding, multiplying, rationalizing (simple and binomial), and FOIL expansion as required by College Board for AP and SAT exams.
Can you add √2 and √3?
No. √2 and √3 are unlike radicals — they have different radicands. The expression √2 + √3 is already in its simplest form. Writing √2 + √3 = √5 is a common mistake.
What is the difference between radicals and surds?
"Radicals" (US term) and "surds" (UK term) refer to the same concept — irrational roots like √2 and √3. The US curriculum uses "radicals" while the UK curriculum uses "surds".
How do I know when a radical is fully simplified?
A radical √n is fully simplified when n has no perfect square factors other than 1. Test: is n divisible by 4, 9, 16, 25, or 36? If yes, simplify further.
Why does multiplying conjugates eliminate the radical?
It uses the algebraic identity (a+b)(a-b) = a²-b². When applied to radicals: (p+√q)(p-√q) = p²-(√q)² = p²-q. Both terms are rational, so the radical vanishes.
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