Simplify Radicals Calculator — Simplest Radical Form with Steps

Algebra Tool

Simplify Radicals Calculator

Q: What is simplest radical form?

A radical expression is in simplest radical form when three conditions are met: (1) no perfect square factors remain under the radical sign other than 1, (2) no fractions appear under the radical sign, and (3) no radicals appear in the denominator (the denominator is rationalized). For example, √72 is NOT in simplest radical form because 72 = 36×2, and 36 is a perfect square. The simplest radical form is 6√2.

Q: How do you simplify a square root?

To simplify a square root: (1) Find the largest perfect square factor of the number under the radical. (2) Write the number as (perfect square) × (remaining factor). (3) Apply the product rule: √(a×b) = √a × √b. (4) Simplify the square root of the perfect square. Example: √72 = √(36×2) = √36 × √2 = 6√2. You can also use prime factorization: 72 = 2³×3², so √72 = √(2²×2×3²) = 2×3×√2 = 6√2.

Q: Can you add √2 and √3?

No, √2 and √3 cannot be combined. Only like radicals — radicals with the same radicand AND the same index — can be added or subtracted. √2 + √3 ≠ √5. This is one of the most common mistakes with radicals. The rule √(a+b) ≠ √a + √b. However, 3√2 + 5√2 = 8√2 because both terms have the same radicand (2). Always simplify each radical first before deciding if they can be combined.

Q: What is the product rule for radicals?

The product rule for radicals states: √a × √b = √(ab), when both a and b are non-negative. This means you can multiply the radicands together. For example: √6 × √10 = √60 = √(4×15) = 2√15. The product rule also works in reverse for simplification. Important: the product rule only works when the indices are the same — you cannot directly multiply √a × ∛b using this rule.

Q: How do you rationalize the denominator?

Rationalizing the denominator means eliminating radicals from the denominator. For a simple radical denominator: multiply both numerator and denominator by the radical. Example: 5/√3 = 5√3/3. For a binomial denominator (a + √b): multiply by the conjugate (a − √b). Example: 4/(3+√5) × (3−√5)/(3−√5) = 4(3−√5)/(9−5) = 4(3−√5)/4 = 3−√5. The conjugate method uses the difference of squares pattern: (a+√b)(a−√b) = a²−b.

Q: What is an extraneous solution in a radical equation?

An extraneous solution is a value that satisfies the equation after squaring both sides but does NOT satisfy the original radical equation. It arises because squaring both sides can introduce false solutions. Example: solving √(x+4) = x−2 gives x=0 or x=5. Check x=0: √4=2 but 0−2=−2, so 2≠−2 — x=0 is extraneous. Check x=5: √9=3 and 5−2=3 ✓. Always substitute ALL solutions back into the ORIGINAL equation to eliminate extraneous solutions.

Q: How do you simplify radicals with variables?

To simplify radicals with variables: for √(xⁿ), bring out x raised to the floor of n/2, and leave x raised to n mod 2 under the radical. Even powers come out completely: √(x⁴) = x². Odd powers leave one factor inside: √(x⁵) = x²√x. For mixed expressions like √(72x³y⁵): simplify the coefficient (√72=6√2), then each variable (√(x³)=x√x, √(y⁵)=y²√y), then combine: 6xy²√(2xy). Assume all variables represent non-negative numbers.

This simplify radicals calculator reduces any radical expression to its simplest radical form, adds and subtracts like radicals, multiplies radicals using the product rule, rationalizes the denominator, and solves radical equations — showing every step from prime factorization to the final simplified form.

Simplify Radicals Calculator — All Tools

Enter the number under the radical (radicand):

Index (root)

√12 → 2√3

√50 → 5√2

√72 → 6√2

√98 → 7√2

√200 → 10√2

√300 → 10√3

√180 → 6√5

√48 → 4√3

∛54 → 3∛2

⁴√80 → 2⁴√5

Simplest Radical Form

Step-by-Step Working

Perfect Square Reference

Expression under the radical (e.g. 72x^3y^5, 50a^4b^7, x^6y^3):

√(8x²)→2x√2

√(18a³)→3a√(2a)

√(45x²y³)→3xy√(5y)

√(72x³y⁵)→6xy²√(2xy)

√(x⁶y²)→x³y

Simplified Radical Expression

Step-by-Step Working

Enter radical terms to add or subtract. Each row: coefficient × √radicand.

3√12+2√27

√50−2√8

√8+√18+√5

2√12+3√27−√75

√75+√48

Combined Result

Step-by-Step Working

Coefficient 1

Radicand 1

Coefficient 2

Radicand 2

3√8 × 5√6

4√6 × 3√10

2√15 × 3√5

√2 × √8

Result

Step-by-Step Working

Solve equations of the form √(ax + b) = c

a (coefficient of x)

b (constant inside)

c (right side)

      √(2x + (−3)) = 5
    

√(2x−3)=5

√(3x+1)=4

√(x+4)=3

√x=7

√(x+9)=−3 (no soln)

Solution

Step-by-Step Working

What Is Simplest Radical Form? — Definition and Rules

A radical expression is in its simplest radical form when it satisfies all three of the following conditions. These are mathematical conventions that create a standardized form making it easy to compare, add, and work with radical expressions.

Condition 1: No Perfect Square Factors Under the Radical

No perfect square factor (other than 1) may remain under the radical sign. Perfect squares are 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144... When you see one of these as a factor of the radicand, you can pull its square root outside.

Example violation: √72 is NOT in simplest radical form — 72 = 36 × 2 and 36 is a perfect square. The simplest radical form is 6√2, because we pull √36 = 6 outside the radical.

Condition 2: No Fractions Under the Radical

No fraction may appear inside the radical sign. If you have √(3/4), rewrite it as √3/√4 = √3/2. This uses the quotient rule for radicals in reverse to eliminate the fraction inside.

Example violation: √(1/2) is NOT in simplest radical form. Correct form: √2/2 (rationalize the denominator).

Condition 3: No Radicals in the Denominator (Rationalize the Denominator)

No radical may appear in the denominator of a fraction. If you have 1/√2, you must rationalize the denominator by multiplying by √2/√2 to get √2/2. When the denominator is a binomial like (3 + √5), multiply by the conjugate (3 − √5).

Example violation: 5/√3 is NOT in simplest radical form. Correct form: 5√3/3 (multiply numerator and denominator by √3).

How to Simplify Radicals — Step-by-Step Method

There are two reliable methods to simplify radicals: the largest perfect square method (faster) and the prime factorization method (more systematic for large numbers). Both always give the same result.

Method 1: Largest Perfect Square Factor (Fastest)

List the factor pairs of the radicand
Identify all perfect square factors: 4, 9, 16, 25, 36, 49, 64, 81, 100…
Select the largest perfect square factor
Write the radicand as (largest perfect square) × (remaining)
Apply the product rule: √(a×b) = √a × √b
Simplify the square root of the perfect square

Method 2: Prime Factorization (Systematic)

Find the complete prime factorization of the radicand
Group prime factors into pairs
Each pair contributes one factor outside the radical
Unpaired factors remain inside the radical

Example: Simplify √48 — Both Methods

Method 1 (Largest Perfect Square):

Perfect square factors of 48: 4, 16. Largest = 16
48 = 16 × 3
√48 = √(16 × 3) = √16 × √3 = 4√3

Method 2 (Prime Factorization):

48 = 2 × 24 = 2 × 2 × 12 = 2 × 2 × 2 × 6 = 2⁴ × 3
√48 = √(2⁴ × 3) = √(2² × 2² × 3)
Each pair of 2s contributes one 2 outside: 2 × 2 × √3 = 4√3
Result: 4√3 — same answer ✓

Example: Simplify √180

Method 1: 180 = 36 × 5 → √180 = 6√5

Method 2: 180 = 2² × 3² × 5 → √180 = 2 × 3 × √5 = 6√5 ✓

Example: Simplify √200

Method 1: 200 = 100 × 2 → √200 = 10√2

Method 2: 200 = 2³ × 5² → √200 = 2 × 5 × √2 = 10√2 ✓

Key insight: If you accidentally use a smaller perfect square (e.g., pulling out 4 from 72 to get 2√18), the answer is NOT fully simplified. Always find the LARGEST perfect square factor. Check if what remains under the radical still has a perfect square factor.

Simplifying Radical Expressions with Variables

For variables under a radical, the rule is: bring out the largest power that results in a whole number after dividing by the root index. For simplify radicals with a square root (index 2):

√(xⁿ) = x^⌊n/2⌋ · √(x^(n mod 2)) Even powers come out completely; odd powers leave one factor under the radical

Under Radical	Simplification	Reasoning
√(x²)	\|x\| (or x if x≥0)	x² is a perfect square
√(x⁴)	x²	x⁴ = (x²)², even power
√(x⁵)	x²√x	x⁵ = x⁴·x, bring out x²
√(x³)	x√x	x³ = x²·x, bring out x
√(x³y⁴)	xy²√x	x³=x²·x; y⁴=(y²)²
√(x⁶y²)	x³y	Both perfect squares — no radical needed

Full Example: Simplify √(72x³y⁵)

Separate: √72 × √(x³) × √(y⁵)
Simplify √72: 72 = 36×2, so √72 = 6√2
Simplify √(x³): x³ = x²·x, so √(x³) = x√x
Simplify √(y⁵): y⁵ = y⁴·y, so √(y⁵) = y²√y
Outside parts: 6 × x × y² = 6xy²
Inside parts: √2 × √x × √y = √(2xy)
Result: 6xy²√(2xy) (assuming x≥0, y≥0)

Domain Warning: √(x²) = |x|, NOT x, when x could be negative. We write √(x²) = x only when we know x ≥ 0. This is the most commonly missed domain restriction when simplifying radicals with variables.

Adding and Subtracting Radical Expressions

Only like radicals — radicals with the same radicand AND the same index — can be combined by addition or subtraction. Like radicals behave just like like terms: a√n + b√n = (a+b)√n.

Critical rule: √(a+b) ≠ √a + √b — this is the single most common mistake with radicals. You CANNOT distribute the radical over addition. √9 + √16 = 3 + 4 = 7, but √(9+16) = √25 = 5 ≠ 7.

Critical first step: Always simplify radicals before deciding if they are like radicals. Many radicals that look unlike become like after simplification.

Example 1 (Simple): 3√5 + 2√5

Both terms already simplified. Same radicand (5) → like radicals ✓
(3+2)√5 = 5√5

Example 2 (Simplify First): √12 + √27

Simplify each: √12 = 2√3, √27 = 3√3
Now both have radicand 3 → like radicals ✓
2√3 + 3√3 = 5√3

Example 3 (Unlike Radicals): √8 + √18 + √5

Simplify each: √8 = 2√2, √18 = 3√2, √5 = √5
Group by radicand: (2√2 + 3√2) + √5
Combine like radicals: 5√2 + √5
Result: 5√2 + √5 — cannot simplify further (different radicands)

Multiplying Radicals — The Product Rule

The product rule for radicals: √a × √b = √(ab), provided both a and b are non-negative. This product rule √a × √b = √(ab) works in both directions — use it to multiply radicals or to split a radical for simplification. The product rule √a × √b = √(ab) is the foundation of all radical multiplication.

√a × √b = √(ab) | a√m × b√n = (ab)√(mn) Product rule for radicals — same index only. Then simplify the result using the product rule √a × √b = √(ab).

Example: 3√8 × 5√6

Multiply coefficients: 3 × 5 = 15
Apply product rule: √8 × √6 = √48 (product rule √a × √b = √(ab))
Combined: 15√48
Simplify √48 = 4√3
Result: 15 × 4√3 = 60√3

Example: FOIL — (2 + √3)(5 − √3)

First: 2×5 = 10
Outer: 2×(−√3) = −2√3
Inner: √3×5 = 5√3
Last: √3×(−√3) = −(√3)² = −3
Combine: 10 − 2√3 + 5√3 − 3 = 7 + 3√3
Result: 7 + 3√3

Important: The product rule for radicals only works when indices are the same. You cannot multiply √a × ∛b directly using √a × √b = √(ab) — they have different indices (2 and 3). Convert to rational exponents first: a^(1/2) × b^(1/3).

Rationalizing the Denominator — Simple and Conjugate Methods

The convention of not leaving radicals in the denominator is called rationalizing the denominator. It creates a standard form and simplifies further arithmetic.

Simple Case: a/√b

Multiply both numerator and denominator by √b. Since √b/√b = 1, the value is unchanged. Result: a√b/b.

Example: Rationalize 5/√3

Multiply by √3/√3: (5/√3) × (√3/√3)
Numerator: 5√3
Denominator: √3 × √3 = 3
Result: 5√3/3

Conjugate Case: c/(a ± √b)

When the denominator is a binomial containing a radical, multiply by the conjugate. The conjugate of (a + √b) is (a − √b). The product uses the difference of squares: (a + √b)(a − √b) = a² − b, which eliminates the radical from the denominator.

Example: Rationalize 4/(3 + √5)

Conjugate of (3 + √5) is (3 − √5)
Multiply: 4/(3+√5) × (3−√5)/(3−√5)
Numerator: 4(3−√5) = 12 − 4√5
Denominator: (3+√5)(3−√5) = 3² − (√5)² = 9 − 5 = 4
Result: (12 − 4√5)/4
Simplify: divide both terms by 4 → 3 − √5

Example: Rationalize 3/(2 − √7)

Conjugate of (2 − √7) is (2 + √7)
Multiply: 3/(2−√7) × (2+√7)/(2+√7)
Numerator: 3(2+√7) = 6 + 3√7
Denominator: 2² − (√7)² = 4 − 7 = −3
Result: (6 + 3√7)/(−3) = −2 − √7

Solving Radical Equations — Isolate and Square

A radical equation contains a variable under a radical sign. The strategy: isolate the radical, then square both sides. But squaring both sides can introduce extraneous solutions — the CHECK step is mandatory, not optional.

Three-Step Method

Isolate the radical on one side of the equation
Square both sides to eliminate the radical (if √(expression) = c, then expression = c²)
CHECK every solution by substituting back into the ORIGINAL equation

Example: Solve √(2x − 3) = 5

Right side c = 5 ≥ 0 ✓ (solution may exist)
Square both sides: 2x − 3 = 25
Solve: 2x = 28 → x = 14
CHECK (CRITICAL): √(2(14) − 3) = √(28 − 3) = √25 = 5 = right side ✓
Solution: x = 14

Extraneous solution warning: When you square both sides of a radical equation, you may introduce solutions that satisfy the squared equation but NOT the original equation. These are called extraneous solutions and must be rejected. ALWAYS substitute every candidate solution back into the ORIGINAL equation to check.

Example with Extraneous Solution: √(x + 4) = x − 2

Square both sides: x + 4 = (x−2)² = x² − 4x + 4
Rearrange: 0 = x² − 5x → x(x−5) = 0
Solutions: x = 0 or x = 5
CHECK x = 0: √(0+4) = √4 = 2, but 0−2 = −2. Since 2 ≠ −2, x = 0 is EXTRANEOUS ✗
CHECK x = 5: √(5+4) = √9 = 3, and 5−2 = 3. Since 3 = 3 ✓, x = 5 is valid.
Solution: x = 5 only — reject x = 0 (extraneous solution)

Common Mistakes With Radical Simplification

Mistake 1 — Adding Unlike Radicals: √2 + √3 ≠ √5

❌ Wrong: √2 + √3 = √5
✅ Correct: √2 + √3 cannot be combined — different radicands make them unlike radicals
Remember: √(a+b) ≠ √a + √b — this rule ONLY works for multiplication/division, not addition

Mistake 2 — Incomplete Simplification

❌ Wrong: √72 = 2√18 (pulled out 4, not 36)
✅ Correct: √72 = 6√2 (must use the LARGEST perfect square factor, which is 36)
Check: does √18 have a perfect square factor? Yes — 18 = 9×2, so 2√18 = 2×3√2 = 6√2

Mistake 3 — Skipping the Check in Radical Equations

❌ Wrong: Accepting all solutions from the squared equation without checking
✅ Correct: Always substitute every candidate solution into the ORIGINAL radical equation
Extraneous solutions arise because (−5)² = 25 = 5², so squaring masks sign information

Mistake 4 — Product Rule Applied to Addition: √(a+b) ≠ √a + √b

❌ Wrong: √(4+9) = √4 + √9 = 2 + 3 = 5
✅ Correct: √(4+9) = √13 ≈ 3.606 (not 5)
The product rule √a × √b = √(ab) works ONLY for multiplication, never addition

Mistake 5 — Forgetting |x| When Simplifying √(x²)

❌ Wrong: √(x²) = x (fails when x is negative)
✅ Correct: √(x²) = |x| in general; √(x²) = x ONLY when x ≥ 0
Example: if x = −3, then x² = 9, √9 = 3 = |−3|, NOT −3

Worked Examples — 10 Complete Problems

1. Simplify √108

108 = 36 × 3 (36 is the largest perfect square factor)
√108 = √36 × √3 = 6√3
Prime check: 108 = 2²×3³ → √108 = 2×3√3 = 6√3 ✓
Answer: 6√3 · Decimal: √108 ≈ 10.392 = 6×1.732 ✓

2. Simplify √(50x⁴y³)

√50 = 5√2 (50 = 25×2)
√(x⁴) = x² (even power, perfect square)
√(y³) = y√y (y³ = y²×y)
Outside: 5×x²×y = 5x²y · Inside: √(2y)
Answer: 5x²y√(2y)

3. Add: 2√12 + 3√27

2√12 = 2×2√3 = 4√3
3√27 = 3×3√3 = 9√3
Like radicals (radicand 3): 4√3 + 9√3 = 13√3
Answer: 13√3

4. Subtract: √50 − 2√8

√50 = 5√2, 2√8 = 2×2√2 = 4√2
5√2 − 4√2 = √2
Answer: √2

5. Add Unlike: √8 + √18 + √3

√8 = 2√2, √18 = 3√2, √3 = √3
Like groups: 2√2 + 3√2 = 5√2 and √3 (unlike — different radicand)
Answer: 5√2 + √3

6. Multiply: 4√6 × 3√10

Coefficients: 4×3=12; Radicands: √6×√10=√60 (product rule √a×√b=√(ab))
Simplify √60: 60=4×15, √60=2√15
12×2√15 = 24√15

7. FOIL: (2 + √3)(5 − √3)

F: 2×5=10 · O: 2×(−√3)=−2√3 · I: √3×5=5√3 · L: √3×(−√3)=−3
Combine: (10−3) + (−2√3+5√3) = 7+3√3
Answer: 7 + 3√3

8. Rationalize: 6/√12

Multiply by √12/√12: 6√12/12 = √12/2
Simplify √12 = 2√3: (2√3)/2 = √3
Answer: √3

9. Rationalize Denominator: 3/(2 + √7)

Conjugate of (2+√7) is (2−√7)
Numerator: 3(2−√7) = 6−3√7
Denominator: 4−7=−3
(6−3√7)/(−3) = −2+√7
Answer: √7 − 2

10. Solve: √(3x + 1) = 4, check for extraneous solutions

c=4≥0 ✓
Square: 3x+1=16 → 3x=15 → x=5
CHECK: √(3(5)+1)=√16=4=right side ✓
Solution: x=5 (valid, no extraneous solution)

Frequently Asked Questions

What is simplest radical form?

Simplest radical form requires three conditions: (1) no perfect square factors remain under the radical, (2) no fractions under the radical, (3) no radicals in the denominator. Example: √72 must be simplified to 6√2 because 72 = 36×2 and 36 is a perfect square factor. Use this simplify radicals calculator to check any expression instantly.

How do you simplify a square root?

To simplify a square root: find the largest perfect square factor of the radicand, write it as (perfect square)×(remainder), apply the product rule √(ab)=√a×√b, then simplify √(perfect square). Example: √72=√(36×2)=6√2. Alternatively, use prime factorization and group prime factors in pairs.

Can you add √2 and √3?

No. √2 and √3 are unlike radicals with different radicands, so they cannot be combined. √2 + √3 ≠ √5. The rule √(a+b) ≠ √a + √b — radicals do not add like regular numbers. Only like radicals (same radicand AND same index) can be added: 3√2 + 5√2 = 8√2.

What is the product rule for radicals?

The product rule for radicals states: √a × √b = √(ab), for non-negative a and b. This product rule allows multiplying radicands directly. Example: √6 × √10 = √60 = 2√15. The product rule √a × √b = √(ab) only works when both radicals have the same index.

How do you rationalize the denominator?

For a/√b: multiply numerator and denominator by √b to get a√b/b. For c/(a±√b): multiply by the conjugate (a∓√b)/(a∓√b). The conjugate method uses (a+√b)(a−√b)=a²−b (difference of squares), which eliminates the radical from the denominator entirely.

What is an extraneous solution in a radical equation?

An extraneous solution satisfies the equation after squaring but not the original radical equation. Example: solving √(x+4)=x−2 yields x=0 and x=5. Checking x=0: √4=2 but 0−2=−2 → 2≠−2, so x=0 is extraneous. Checking x=5: √9=3=5−2=3 ✓. Always check ALL solutions in the ORIGINAL equation.

How do you simplify radicals with variables?

For √(xⁿ): bring out x^⌊n/2⌋ and leave x^(n mod 2) under the radical. Even powers come out completely (√(x⁴)=x²); odd powers leave one factor inside (√(x⁵)=x²√x). For mixed expressions like √(72x³y⁵), simplify the numeric coefficient (6√2) and each variable separately, then combine: 6xy²√(2xy). Assume all variables are non-negative.

Related Calculators

Quick Rules

√(ab) = √a × √bProduct rule — same index only

√xⁿ = x^⌊n/2⌋ · √(x^(n%2))Variable simplification rule

a√n + b√n = (a+b)√nLike radicals — combine coefficients

√a+√b ≠ √(a+b)Unlike radicals — cannot add!

a/√b = a√b/bRationalize simple denominator

(a+√b)(a−√b) = a²−bConjugate / difference of squares

√(x²) = |x|Always |x|, not just x

Perfect Squares

√4 = 2

√9 = 3

√16 = 4

√25 = 5

√36 = 6

√49 = 7

√64 = 8

√81 = 9

√100 = 10

√121 = 11

√144 = 12

√169 = 13

√196 = 14

√225 = 15

√256 = 16

√400 = 20

Most Searched

√72 = 6√2

√50 = 5√2

√48 = 4√3

√98 = 7√2

√200 = 10√2

√180 = 6√5

√12 = 2√3

√300 = 10√3

√75 = 5√3

√108 = 6√3

Related Tools

Simplify Radicals Calculator — Simplest Radical Form with Steps

Simplify Radicals Calculator

What Is Simplest Radical Form? — Definition and Rules

Condition 1: No Perfect Square Factors Under the Radical

Condition 2: No Fractions Under the Radical

Condition 3: No Radicals in the Denominator (Rationalize the Denominator)

How to Simplify Radicals — Step-by-Step Method

Method 1: Largest Perfect Square Factor (Fastest)

Method 2: Prime Factorization (Systematic)

Example: Simplify √48 — Both Methods

Example: Simplify √180

Example: Simplify √200

Simplifying Radical Expressions with Variables

Full Example: Simplify √(72x³y⁵)

Adding and Subtracting Radical Expressions

Example 1 (Simple): 3√5 + 2√5

Example 2 (Simplify First): √12 + √27

Example 3 (Unlike Radicals): √8 + √18 + √5

Multiplying Radicals — The Product Rule

Example: 3√8 × 5√6

Example: FOIL — (2 + √3)(5 − √3)

Rationalizing the Denominator — Simple and Conjugate Methods

Simple Case: a/√b

Example: Rationalize 5/√3

Conjugate Case: c/(a ± √b)

Example: Rationalize 4/(3 + √5)

Example: Rationalize 3/(2 − √7)

Solving Radical Equations — Isolate and Square

Three-Step Method

Example: Solve √(2x − 3) = 5

Example with Extraneous Solution: √(x + 4) = x − 2

Common Mistakes With Radical Simplification

Mistake 1 — Adding Unlike Radicals: √2 + √3 ≠ √5

Mistake 2 — Incomplete Simplification

Mistake 3 — Skipping the Check in Radical Equations

Mistake 4 — Product Rule Applied to Addition: √(a+b) ≠ √a + √b

Mistake 5 — Forgetting |x| When Simplifying √(x²)

Worked Examples — 10 Complete Problems

1. Simplify √108

2. Simplify √(50x⁴y³)

3. Add: 2√12 + 3√27

4. Subtract: √50 − 2√8

5. Add Unlike: √8 + √18 + √3

6. Multiply: 4√6 × 3√10

7. FOIL: (2 + √3)(5 − √3)

8. Rationalize: 6/√12

9. Rationalize Denominator: 3/(2 + √7)

10. Solve: √(3x + 1) = 4, check for extraneous solutions

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