Square Root Calculator

Square Root Calculator

Calculate square roots instantly with our free online calculator. Find √ of any number with high precision, simplify radicals, and get step-by-step explanations.

Calculate Square Root

Enter a Number: [Input field: e.g., 144]

Precision: [Dropdown: 2, 4, 6, 8, 10 decimal places]

[Calculate Button]

Results:

• Square Root: √[n] = [Result]
• Decimal Form: [Result]
• Perfect Square: [Yes/No]
• Radical Form (if applicable): [Simplified]

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What is a Square Root?

A square root of a number is a value that, when multiplied by itself, gives the original number. Every positive number has two square roots: one positive and one negative.

Basic Definition

If x² = n Then x = √n

Example:

5 × 5 = 25
Therefore: √25 = 5

Square Root Symbol

The symbol √ is called the radical sign.

Anatomy of √:

√n
│└─ radicand (the number under the root)
└─ index (omitted for square root, understood as 2)

Why Square Roots Matter

1. Geometry: Calculate diagonal of squares, side length from area
2. Physics: Formulas for velocity, energy, waves
3. Engineering: Structural calculations, signal processing
4. Finance: Standard deviation, risk calculations
5. Daily Life: Screen sizes, construction, scaling

Perfect Squares

List of Perfect Squares (1-25)

1² = 1     →  √1 = 1
2² = 4     →  √4 = 2
3² = 9     →  √9 = 3
4² = 16    →  √16 = 4
5² = 25    →  √25 = 5
6² = 36    →  √36 = 6
7² = 49    →  √49 = 7
8² = 64    →  √64 = 8
9² = 81    →  √81 = 9
10² = 100  →  √100 = 10
11² = 121  →  √121 = 11
12² = 144  →  √144 = 12
13² = 169  →  √169 = 13
14² = 196  →  √196 = 14
15² = 225  →  √225 = 15
16² = 256  →  √256 = 16
17² = 289  →  √289 = 17
18² = 324  →  √324 = 18
19² = 361  →  √361 = 19
20² = 400  →  √400 = 20
21² = 441  →  √441 = 21
22² = 484  →  √484 = 22
23² = 529  →  √529 = 23
24² = 576  →  √576 = 24
25² = 625  →  √625 = 25

Identifying Perfect Squares

Quick checks:

• Ends in: 0, 1, 4, 5, 6, or 9
• Digital root is 1, 4, 7, or 9

Examples:

• 144: Ends in 4, could be perfect square ✓
• 150: Ends in 0, could be perfect square ✓
• 147: Ends in 7, NOT a perfect square ✗

How to Calculate Square Roots

Method 1: Using Perfect Squares

Best for: Numbers that are perfect squares

Example: √225

Find: 15 × 15 = 225
Therefore: √225 = 15

Method 2: Prime Factorization

Best for: Simplifying radicals, finding exact forms

Example: √180

Step 1: Prime factorize

180 = 2² × 3² × 5

Step 2: Extract perfect squares

√180 = √(2² × 3² × 5)
√180 = √(2²) × √(3²) × √(5)
√180 = 2 × 3 × √5
√180 = 6√5

Decimal: √180 ≈ 13.416

Method 3: Estimation and Refinement

Best for: Non-perfect squares, mental calculation

Example: √50

Step 1: Find closest perfect squares

√49 = 7
√64 = 8
√50 is between 7 and 8

Step 2: Estimate

50 is closer to 49 than 64
Estimate: ≈ 7.1

Step 3: Verify and refine

7.1² = 50.41 (too high)
7.07² ≈ 49.98 (very close)
√50 ≈ 7.071

Method 4: Long Division Method

Best for: Calculating square roots by hand

Example: √529

     2  3
   ┌─────
 2 │ 5 29
   │ 4
   └────
     1 29
   4 3│ 1 29
      │ 1 29
      └─────
        0

Therefore: √529 = 23

Square Root Examples

Example 1: Perfect Square

Calculate: √196

Solution:

14 × 14 = 196
√196 = 14

Example 2: Non-Perfect Square

Calculate: √72

Solution:

Prime factorization: 72 = 2³ × 3²
√72 = √(2² × 2 × 3²)
√72 = 2 × 3 × √2
√72 = 6√2

Decimal: √72 ≈ 8.485

Example 3: Large Number

Calculate: √2025

Solution:

40² = 1600
50² = 2500
√2025 is between 40 and 50
Try 45: 45² = 2025 ✓
√2025 = 45

Example 4: Decimal Number

Calculate: √0.25

Solution:

0.5 × 0.5 = 0.25
√0.25 = 0.5

Example 5: Fraction

Calculate: √(16/25)

Solution:

√(16/25) = √16 / √25
√(16/25) = 4/5

Simplifying Square Roots

Rules

1. Extract perfect squares
2. Simplify fraction under root
3. Rationalize denominator (remove roots from denominator)

Examples

Example 1: √50

√50 = √(25 × 2)
√50 = √25 × √2
√50 = 5√2

Example 2: √(18/25)

√(18/25) = √18 / √25
√(18/25) = √(9 × 2) / 5
√(18/25) = (3√2) / 5

Example 3: 1/√2

1/√2 = (1 × √2) / (√2 × √2)
1/√2 = √2 / 2

Properties of Square Roots

1. Product Rule

√(a × b) = √a × √b

Example:

√(16 × 9) = √16 × √9
√144 = 4 × 3
√144 = 12

2. Quotient Rule

√(a/b) = √a / √b

Example:

√(36/4) = √36 / √4
√9 = 6 / 2
√9 = 3

3. Power Rule

√(a²) = |a| (absolute value)

Example:

√(5²) = 5
√((-5)²) = 5

4. Additive Property (No Simplification)

√(a + b) ≠ √a + √b

Example:

√(9 + 16) = √25 = 5
√9 + √16 = 3 + 4 = 7
5 ≠ 7 ✗

Common Square Roots

Frequently Used Values

Number	Square Root	Decimal
2	√2	1.414
3	√3	1.732
5	√5	2.236
6	√6	2.449
7	√7	2.646
10	√10	3.162

Powers of 2

2ⁿ	Value	√(2ⁿ)
2² = 4	√4 = 2
2³ = 8	√8 = 2√2 ≈ 2.828
2⁴ = 16	√16 = 4
2⁵ = 32	√32 = 4√2 ≈ 5.657
2⁶ = 64	√64 = 8

Applications of Square Roots

1. Pythagorean Theorem

Formula: c² = a² + b² Therefore: c = √(a² + b²)

Example: Find hypotenuse of right triangle with legs 3 and 4

c = √(3² + 4²)
c = √(9 + 16)
c = √25
c = 5

2. Area to Side Length

Square: A = s² Therefore: s = √A

Example: Square with area 144 m²

s = √144
s = 12 m

3. Distance Formula

d = √[(x₂ - x₁)² + (y₂ - y₁)²]

Example: Distance between (1, 2) and (4, 6)

d = √[(4 - 1)² + (6 - 2)²]
d = √[3² + 4²]
d = √[9 + 16]
d = √25
d = 5

4. Quadratic Formula

x = (-b ± √(b² - 4ac)) / (2a)

5. Standard Deviation

σ = √[Σ(xᵢ - μ)² / N]

6. Screen Sizes

Diagonal of rectangle:

d = √(width² + height²)

Example: 1920×1080 screen

d = √(1920² + 1080²)
d = √(3,686,400 + 1,166,400)
d = √4,852,800
d ≈ 2203 pixels

Estimating Square Roots

Quick Estimation Method

Example: Estimate √70

Step 1: Find perfect squares

√64 = 8
√81 = 9
√70 is between 8 and 9

Step 2: Interpolate

70 is 6/17 of the way from 64 to 81
Estimate: 8 + (6/17) ≈ 8.35

Step 3: Refine

8.37² ≈ 70.06
√70 ≈ 8.367

Mental Math Tricks

For numbers ending in 25:

√625 = 25
√1225 = 35
√2025 = 45
Pattern: (n)(2n)(25) → (n5)²

Negative Square Roots

Principal vs Negative Root

Every positive number has TWO square roots:

Example: √25

• Principal (positive): +5
• Negative: -5

Notation:

√25 = 5 (principal)
±√25 = ±5 (both)

Imaginary Numbers

√(-1) = i (imaginary unit)

Example:

√(-9) = √(-1 × 9)
√(-9) = √(-1) × √(9)
√(-9) = 3i

Advanced Topics

nth Roots

General form: ⁿ√a

Examples:

• ∛a = cube root
• ⁴√a = fourth root
• ⁵√a = fifth root

Example: ∛27

3 × 3 × 3 = 27
∛27 = 3

Rational Exponents

a^(1/n) = ⁿ√a

Example:

16^(1/2) = √16 = 4
27^(1/3) = ∛27 = 3

Nested Radicals

√(a + √b)

Example: √(6 + √11)

Cannot be simplified further
≈ 2.924

What is a square root?

A square root of a number is a value that multiplied by itself gives the original number. For example, √25 = 5 because 5 × 5 = 25.

How do I calculate square root by hand?

Use methods like: prime factorization (for simplification), estimation and refinement (for decimals), or the long division method (for exact calculation).

What's the square root of 2?

√2 ≈ 1.41421356. It's an irrational number that cannot be expressed as a simple fraction.

Is √(a + b) = √a + √b?

No, this is a common mistake. √(9 + 16) = √25 = 5, but √9 + √16 = 3 + 4 = 7. They're not equal.

Can you take the square root of a negative number?

Square roots of negative numbers involve imaginary numbers. √(-1) = i (the imaginary unit).

What's the difference between √x and ±√x?

√x denotes the principal (positive) square root. ±√x indicates both the positive and negative square roots.

How do I simplify √50?

√50 = √(25 × 2) = 5√2. This is the simplified radical form.

What are perfect squares?

Perfect squares are numbers that are squares of integers: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, etc.

How do I find square root without calculator?

Estimate using perfect squares, use prime factorization for exact forms, or use the long division method.

Why is √(x²) = |x| not x?

Because √ always gives the positive (principal) root. √((-5)²) = √25 = 5, not -5.

What is √0?

√0 = 0 because 0 × 0 = 0.

How are square roots used in real life?

Square roots are used in: construction (diagonal measurements), finance (standard deviation), physics (formulas), computer graphics (distance calculations), and many more fields.

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Practice Problems

Beginner Level

1. √49 = ?
2. √100 = ?
3. √121 = ?
4. √144 = ?
5. √225 = ?

Intermediate Level

1. Simplify: √18
2. Simplify: √75
3. Calculate: √(16/25)
4. Simplify: √32
5. Estimate: √60

Advanced Level

1. Simplify: √72 + √50
2. Solve: x² = 81
3. Calculate: √50 × √2
4. Simplify: (√8)²
5. Find: √(9 + 16)

Answers: [Click to reveal]

1. Beginner: 7, 10, 11, 12, 15
2. Intermediate: 3√2, 5√3, 4/5, 4√2, ≈7.75
3. Advanced: 9√2, x=±9, 10, 8, 5

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• Exponent Calculator
• Scientific Calculator
• Prime Factorization Calculator
• Fraction Calculator
• Percentage Calculator

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