Square Root Calculator

Written by the StepSolvers Team  │  Reviewed by a Math Educator  │  Last updated July 2026

The square root calculator finds the square root of any number — showing the exact radical form, the simplified radical, the prime factorisation, and the decimal approximation, all with step-by-step working. Enter any number below to get an instant result with a full explanation.

This free calculator handles perfect squares and non-perfect squares (simplified to radical form). A separate Cube Root Calculator further down this page finds cube roots. No signup required.

Step-by-Step Solution

    Final Answer:

    What Is a Square Root?

    The square root of a number is the value that, when multiplied by itself, gives that number. For example, the square root of 25 is 5 because 5 × 5 = 25. The square root is written using the radical symbol √. Khan Academy's introduction to square roots is a good next stop for more practice.

    Square root notation: √n = x means x × x = n Where: √ = the radical symbol n = the radicand (the number under the radical) x = the square root (the result) Example: √49 = 7 because 7 × 7 = 49 Example: √2 ≈ 1.414 (irrational — cannot be expressed as a fraction)

    Perfect Squares vs Non-Perfect Squares

    TypeDefinition and examples
    Perfect squareA number whose square root is an exact integer. Examples: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225.
    Non-perfect squareA number whose square root is irrational — it cannot be expressed as a fraction and its decimal goes on forever. Examples: √2, √3, √5, √7, √10.
    Simplified radicalA non-perfect square written in its simplest radical form. Example: √72 = 6√2 (6 times the square root of 2).
    Principal square rootThe positive square root of a number. By convention, √n always refers to the positive root. Example: √25 = 5, not −5.

    The Square Root Symbol (√)

    The radical symbol √ is used to denote the square root. The number inside is called the radicand. The small number written to the upper-left of the radical indicates the root — for square roots this is 2 (usually omitted), and for cube roots it is 3 (written as ³√), which the Cube Root Calculator further down this page handles directly.

    Square Root Formula and Methods

    √n = n^(1/2) ← square root is the same as raising to the power of 1/2 For simplifying: √(a × b) = √a × √b For fractions: √(a/b) = √a / √b The key identity for simplification: √(p² × q) = p√q where p is the largest perfect square factor of the radicand

    Method 1: Prime Factorisation (exact simplified form)

    Prime factorisation is the most reliable method for simplifying any square root. Factor the number into its prime factors, then pair up identical primes — each pair comes out of the radical as one factor. This is the method the calculator above uses.

    Method 2: Long Division Method (without a calculator)

    The long division method finds the decimal value of any square root without a calculator. It works by grouping digits in pairs from the decimal point and performing a modified division process. This method is taught in Class 8 and Class 9 in most curricula.

    Method 3: Estimation

    For a quick approximation, identify the two perfect squares the number falls between, then interpolate. For example, √50 lies between √49=7 and √64=8. Since 50 is close to 49, √50 ≈ 7.07.

    How to Use This Square Root Calculator

    1. Enter the number in the input field. The calculator accepts positive integers, decimals, and large numbers.
    2. Click Calculate. The result appears with the exact radical form, simplified radical, prime factorisation, and decimal approximation.
    3. The calculator shows whether your number is a perfect square. If it is not, it displays the simplified radical form alongside the decimal.
    4. The prime factorisation tree is shown step by step so you can follow the working.
    5. Click Reset to start a new calculation.
    6. For cube roots, use the separate Cube Root Calculator further down this page.

    Worked Examples — Step by Step

    Example 1: Square Root of a Perfect Square — √144

    1. Check if 144 is a perfect square: is there an integer n where n×n = 144?
    2. Test: 12 × 12 = 144 ✓ → 144 is a perfect square
    3. Prime factorisation: 144 = 2 × 72 = 2 × 2 × 36 = 2 × 2 × 4 × 9 = 2⁴ × 3²
    4. Group into pairs: (2²) × (2²) × (3²)
    5. Take one from each pair: √144 = 2 × 2 × 3 = 12
    6. Result: √144 = 12 (exact — no decimal needed)
    √144 = 12 (perfect square)

    Example 2: Simplify a Non-Perfect Square — √72

    √72 is not a perfect square, but it can be simplified to radical form using prime factorisation.

    1. Find the prime factorisation of 72: 72 = 2 × 36 = 2 × 4 × 9 = 2³ × 3²
    2. Rewrite: √72 = √(2³ × 3²) = √(2² × 2 × 3²)
    3. Group perfect square factors: √(2² × 3² × 2)
    4. Apply √(a² × b) = a√b for each perfect square pair
    5. √(2²) = 2 and √(3²) = 3 → 2 × 3 = 6
    6. Remaining factor under radical: √2
    7. Final simplified form: √72 = 6√2
    8. Decimal: 6 × 1.4142... = 8.485...
    √72 = 6√2 ≈ 8.485

    Example 3: How to Find Square Root Without a Calculator — √2 by Estimation

    Using the estimation method to find √2 to 3 decimal places.

    1. Identify surrounding perfect squares: √1 = 1 and √4 = 2, so √2 is between 1 and 2
    2. First estimate: try 1.5 → 1.5² = 2.25 (too high)
    3. Narrow down: try 1.4 → 1.4² = 1.96 (too low)
    4. Narrow down: try 1.41 → 1.41² = 1.9881 (close, slightly low)
    5. Narrow down: try 1.414 → 1.414² = 1.999396 (very close)
    6. Narrow down: try 1.4142 → 1.4142² = 1.99996164 (accurate to 4 decimal places)
    7. Result: √2 ≈ 1.4142 (irrational — decimal continues forever without repeating)
    √2 ≈ 1.4142 (irrational)

    Example 4: Cube Root — ³√216

    Try this one in the Cube Root Calculator below to check the working.

    1. Find the cube root of 216: what number multiplied by itself 3 times gives 216?
    2. Prime factorisation: 216 = 2 × 108 = 2 × 2 × 54 = 2 × 2 × 2 × 27 = 2³ × 3³
    3. Group into triples: (2 × 2 × 2) × (3 × 3 × 3)
    4. Take one from each triple: ³√216 = 2 × 3 = 6
    5. Verify: 6 × 6 × 6 = 216 ✓
    ³√216 = 6 (perfect cube)

    Square Root Table 1 to 30

    This reference table shows the square roots of the first 30 integers, with the simplified radical form for non-perfect squares and the decimal approximation to 3 places.

    NumberSquare RootSimplifiedDecimalType
    1√111.000Perfect square
    2√2√21.414Irrational
    3√3√31.732Irrational
    4√422.000Perfect square
    5√5√52.236Irrational
    6√6√62.449Irrational
    7√7√72.646Irrational
    8√82√22.828Irrational
    9√933.000Perfect square
    10√10√103.162Irrational
    12√122√33.464Irrational
    16√1644.000Perfect square
    18√183√24.243Irrational
    20√202√54.472Irrational
    25√2555.000Perfect square
    27√273√35.196Irrational
    36√3666.000Perfect square
    49√4977.000Perfect square
    50√505√27.071Irrational
    64√6488.000Perfect square
    72√726√28.485Irrational
    81√8199.000Perfect square
    100√1001010.000Perfect square
    Perfect squares from 1 to 100: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 Perfect squares from 100 to 225: 121, 144, 169, 196, 225

    How to Simplify Square Roots

    Simplifying a square root means writing it in its simplest radical form — where no perfect square factors remain under the radical. The process uses prime factorisation to identify and extract perfect square factors.

    Process for simplifying √n: 1. Find the prime factorisation of n 2. Group the prime factors into pairs 3. For each pair, move one factor outside the radical 4. Any unpaired factors remain under the radical Examples: √50 = √(25 × 2) = √25 × √2 = 5√2 √98 = √(49 × 2) = √49 × √2 = 7√2 √200 = √(2³ × 5²) = √(2² × 2 × 5²) = 2 × 5 × √2 = 10√2

    Root Mean Square (RMS)

    The root mean square is a special type of average used in physics, engineering, and statistics. It is the square root of the mean of the squared values in a dataset — a calculation you can also perform step by step with the Average Calculator. RMS is used for AC electrical calculations, signal processing, and any situation where both positive and negative values occur.

    RMS formula: RMS = √[(x₁² + x₂² + ... + xₙ²) / n] Example: find the RMS of 3, 4, and 5 Step 1: Square each value: 9, 16, 25 Step 2: Find the mean of squares: (9+16+25)/3 = 50/3 = 16.67 Step 3: Take the square root: √16.67 = 4.08 RMS = 4.08

    Common Mistakes When Working with Square Roots

    • Forgetting the negative root exists: every positive number technically has two square roots (positive and negative), but √n by convention always refers to the positive principal root only.
    • Stopping simplification too early: a radical isn't fully simplified until no perfect square factors remain inside — for example, √72 simplified to 2√18 is incomplete, since 18 still has a perfect square factor (9).
    • Adding radicals like regular numbers: √2 + √3 is not √5 — unlike multiplication, square roots can only be combined by addition or subtraction when the radicand is identical (e.g. 2√3 + 5√3 = 7√3).
    • Treating the square root of a negative number as zero or undefined without qualification: it isn't undefined — it's imaginary. √−9 = 3i, not an error, within the complex number system.

    Frequently Asked Questions

    What is the square root of a number?
    The square root of a number n is the value that, when multiplied by itself, gives n. It is written as √n. For example, √25 = 5 because 5 × 5 = 25. Every positive number has two square roots — a positive one and a negative one — but by convention √n always refers to the positive (principal) square root.
    How do you find a square root without a calculator?
    The two main methods are prime factorisation (for exact simplified form) and the estimation method (for a decimal approximation). For prime factorisation: break the number into prime factors, group them in pairs, and move one from each pair outside the radical. For estimation: identify which two perfect squares the number falls between, then narrow down by testing values and squaring them.
    How do you calculate the square root by hand?
    Use the long division method for a decimal answer without a calculator. Group the digits in pairs from the decimal point. Find the largest integer whose square is less than or equal to the first group — this is your first digit. Bring down the next pair and repeat using a modified divisor that includes the digits found so far.
    How do you simplify a square root?
    Find the prime factorisation of the number under the radical, group the prime factors into pairs, and move one factor outside the radical for each pair. Multiply the factors that came out to get the coefficient. Any unpaired prime factors remain under the radical. For example: √72 = √(2³ × 3²) = √(2² × 3² × 2) = 6√2.
    What is the square root of 2?
    The square root of 2 (√2) is approximately 1.41421356... It is an irrational number — its decimal expansion goes on forever without repeating. √2 appears frequently in geometry (it is the diagonal of a unit square), in trigonometry (sin 45° = cos 45° = √2/2), and throughout mathematics and physics.
    What is the square root of 3?
    The square root of 3 (√3) is approximately 1.73205080... It is also irrational. √3 commonly appears in equilateral triangle calculations (the height of an equilateral triangle with side 1 is √3/2), in 30-60-90 triangles, and in hexagonal geometry.
    What are perfect squares from 1 to 100?
    The perfect squares from 1 to 100 are: 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100. These are the squares of 1 through 10 respectively. Memorising these 10 values allows you to quickly simplify most common square roots.
    Can the square root of a negative number be calculated?
    Not as a real number. The square root of a negative number produces an imaginary number. For example, √(−4) = 2i, where i is the imaginary unit defined as i = √(−1). For standard calculator purposes, the square root is only defined for non-negative numbers.

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