Simplify Radicals Calculator
Simplify radical expressions by extracting perfect square (or nth-power) factors. Converts √72 → 6√2, ∛54 → 3∛2, and handles variables.
Quick examples:
What Is the Simplify Radicals Calculator?
The Simplify Radicals Calculator reduces any integer radical ⁿ√a to its simplest form by prime-factorising the radicand, extracting groups of n identical prime factors, and showing a full step-by-step working. Supports square roots through 10th roots, negative radicands for odd indices, and displays the decimal approximation.
- ›Prime factorisation: break the radicand into prime factors with their exponents
- ›Extract groups: each complete set of n identical primes becomes one factor outside
- ›Remaining primes that cannot form complete groups stay inside the radical
- ›Root index 2–10: supports square, cube, 4th through 10th roots
Formula
Radical Simplification Rules
General form
ⁿ√(aⁿ × b) = a × ⁿ√b
Square root
√(p² × q) = p√q
Cube root
∛(p³ × q) = p∛q
Product rule
√(a × b) = √a × √b
Quotient rule
√(a/b) = √a / √b
Perfect nth
ⁿ√(aⁿ) = a (integer result)
How to Use
- 1Enter the radicand, any whole number (positive, or negative for odd indices)
- 2Select the root index from the dropdown (2 = square root, 3 = cube root, etc.)
- 3Press Enter or click Simplify Radical
- 4The simplified form, coefficient, and decimal approximation are shown
- 5Step-by-step working shows the full prime factorisation and extraction process
- 6Use preset buttons to quickly load common examples
Example Calculation
Simplify √180:
Square root groups (n=2): extract 2² → coefficient 2; extract 3² → coefficient 3
Remaining inside: 5 (exponent 1, less than 2)
Result: 6√5
Decimal: 6 × √5 = 6 × 2.236... = 13.4164...
Simplify ∛54:
Cube root groups (n=3): extract 3³ → coefficient 3
Remaining inside: 2 (exponent 1, less than 3)
Result: 3∛2
Decimal: 3 × ∛2 = 3 × 1.2599... = 3.7798...
When the radical fully simplifies
If all prime exponents are multiples of the root index, the radical evaluates to an integer. √144 = √(2⁴ × 3²) = 2² × 3 = 12. No radical remains, it was a perfect square.
Understanding Simplify Radicals
Perfect Squares and Cubes Reference
| n | n² | n³ | n⁴ | √n² = n | ∛n³ = n |
|---|---|---|---|---|---|
| 1 | 1 | 1 | 1 | √1 = 1 | ∛1 = 1 |
| 2 | 4 | 8 | 16 | √4 = 2 | ∛8 = 2 |
| 3 | 9 | 27 | 81 | √9 = 3 | ∛27 = 3 |
| 4 | 16 | 64 | 256 | √16 = 4 | ∛64 = 4 |
| 5 | 25 | 125 | 625 | √25 = 5 | ∛125 = 5 |
| 6 | 36 | 216 | 1296 | √36 = 6 | ∛216 = 6 |
| 7 | 49 | 343 | 2401 | √49 = 7 | ∛343 = 7 |
| 8 | 64 | 512 | 4096 | √64 = 8 | ∛512 = 8 |
| 9 | 81 | 729 | 6561 | √81 = 9 | ∛729 = 9 |
| 10 | 100 | 1000 | 10000 | √100 = 10 | ∛1000 = 10 |
Frequently Asked Questions
What does it mean to simplify a radical?
Simplifying a radical is the process of extracting the largest perfect-power factor from the radicand, reducing it to the standard form: coefficient × ⁿ√(reduced radicand).
- ›√12 = √(4×3) = 2√3 (4 is a perfect square factor)
- ›√48 = √(16×3) = 4√3 (16 = 4² is the largest perfect-square factor)
- ›Not yet simplified: √48 = 2√12 = 2×2√3 = 4√3 (any form reaching 4√3 is correct)
- ›Fully simplified: radicand has no repeated prime factor ≥ 2
How does the calculator simplify step by step?
The algorithm is exact for integer radicands of any size. It works identically for all root indices, square, cube, 4th, and higher.
- ›Step 1: write radicand = p₁^e₁ × p₂^e₂ × ... (prime factorisation)
- ›Step 2: for each prime pᵢ with exponent eᵢ, compute extracted = floor(eᵢ/n), remaining = eᵢ mod n
- ›Step 3: coefficient = product of pᵢ^extracted; inside = product of pᵢ^remaining
- ›Step 4: result is coefficient × ⁿ√inside
Can you simplify the square root of a negative number?
The sign rule for radicals follows from the definition: ⁿ√(−a) = −ⁿ√a only when n is odd.
- ›∛(−8) = −∛8 = −2 (real result, odd index)
- ›⁵√(−32) = −⁵√32 = −2 (real result, odd index)
- ›√(−4) = 2i (imaginary, even index)
- ›This calculator handles odd-index negative radicands and shows −coefficient × ⁿ√radicand
What is a perfect square, perfect cube, etc.?
Recognizing perfect powers up to common values is a useful mental arithmetic skill for quick simplification without a calculator.
- ›Perfect squares to 400: 1,4,9,16,25,36,49,64,81,100,121,144,169,196,225,256,289,324,361,400
- ›Perfect cubes to 1000: 1,8,27,64,125,216,343,512,729,1000
- ›Perfect 4th powers to 256: 1,16,81,256
- ›Any number ending in 2,3,7,8 cannot be a perfect square
How are radicals used in geometry and trigonometry?
Simplifying radicals is essential for exact answers in geometry, leaving an answer as 6√5 is more precise than the decimal approximation 13.416...
- ›Diagonal of a square with side a: d = a√2 (from Pythagorean theorem)
- ›Height of an equilateral triangle with side a: h = (a√3)/2
- ›30-60-90 triangle sides: 1, √3, 2 (in ratio)
- ›45-45-90 triangle sides: 1, 1, √2 (in ratio)
How do you add or subtract radicals?
Adding radicals is analogous to adding like terms in algebra. The radicand (after simplification) acts like the variable name, you can only combine terms with identical radicands.
- ›√12 + √75 = 2√3 + 5√3 = 7√3 (both simplify to have √3)
- ›√8 + √18 = 2√2 + 3√2 = 5√2
- ›√2 + √3 cannot be simplified further (different radicands)
- ›Always simplify first, then look for matching radicands to combine
What is rationalising the denominator?
Rationalising the denominator is a convention that makes fractions easier to add and compare, it is easier to work with √2/2 than 1/√2, especially when combining fractions.
- ›Simple: 1/√3 × √3/√3 = √3/3
- ›Binomial denominator: 1/(1+√2) × (1−√2)/(1−√2) = (1−√2)/(1−2) = √2−1
- ›Higher root: 1/∛3 = ∛9/3 (multiply by ∛(3²)/∛(3²) = ∛9/3)
- ›Required form on most standardised tests (SAT, ACT, AP exams)