What is a square root?

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 144 is 12 because 12 × 12 = 144. By convention, √x refers to the positive square root (principal square root).

Can you take the square root of a negative number?

Not with real numbers. The square root of a negative number involves imaginary numbers: √(-9) = 3i (where i is the imaginary unit, √-1). This calculator works with non-negative real numbers. Used in complex analysis, AC electronics, quantum mechanics.

How do I simplify √50?

Factor out perfect squares. 50 = 25 × 2. So √50 = √25 × √2 = 5√2. Exact form: 5√2. Decimal: ~7.0711. Always extract perfect square factors when expressing radicals.

What's the difference between square root and cube root?

Square root (√x): value n such that n² = x. Only defined for x ≥ 0 in real numbers. Cube root (∛x): value n such that n³ = x. Defined for all real numbers including negatives (∛-8 = -2). Generalizes: nth root for any positive integer n.

How do I find a square root without a calculator?

Newton's method: start with guess y₀, iterate y_{n+1} = (y_n + x/y_n)/2. For √50: guess 7, get 7.071. Converges quickly. Alternatively: estimate between known perfect squares (√50 is between 7 = √49 and 8 = √64; closer to 7).

How are square roots used in geometry?

Pythagorean theorem: hypotenuse = √(a² + b²). Distance formula: d = √((Δx)² + (Δy)²). Vector magnitude: |v| = √(sum of squares). These appear constantly in geometry, computer graphics, and physics.

What's the square root of 2?

√2 ≈ 1.41421356237... — irrational (non-repeating, non-terminating decimal). Pythagoreans proved this irrationality (~500 BC), shocking ancient mathematicians who believed all numbers were rational. Appears in diagonal of unit square (1×1 square has diagonal √2).

How are radicals different from exponents?

Square root √x = x^(1/2). nth root ⁿ√x = x^(1/n). All roots are fractional exponents. Calculus shows derivatives, integrals work consistently using exponent rules. Equivalent representations: radical or exponent notation.

Square Root Calculator

Find the square root of any positive number, or calculate the nth root by specifying a custom root index. Useful for algebra, geometry, and general math problems.

The square root of a number x, written √x, is the value that when multiplied by itself gives x. So √16 = 4 because 4 × 4 = 16. Square roots are the inverse of squaring — they "undo" exponentiation by 2. They appear throughout mathematics, physics, statistics, and engineering, anywhere quadratic relationships need to be reversed.

Notation: - **√x** (or √(x)): square root (root index 2 implicit). - **∛x**: cube root (root index 3). - **⁴√x**: fourth root, etc. - **x^(1/n)**: equivalent to nth root.

Square roots can be exact or irrational: - **Perfect squares**: √4 = 2, √9 = 3, √16 = 4, √25 = 5, √100 = 10. Exact integer values. - **Irrational**: √2 ≈ 1.41421356..., √3 ≈ 1.732050807..., √7 ≈ 2.6457513.... Decimal expansions never repeat or terminate.

Properties of square roots: - **Non-negative**: √x ≥ 0 always (by convention). - **Of zero**: √0 = 0. - **Of one**: √1 = 1. - **Product**: √(a×b) = √a × √b. - **Quotient**: √(a/b) = √a / √b. - **Of square**: √(x²) = |x| (absolute value).

For negative numbers, there's no real square root. √(-1) = i (imaginary unit), basis of complex numbers.

Common applications: Pythagorean theorem (hypotenuse calculation), distance formula, quadratic formula, standard deviation (statistics), physics (energy formulas, periods), engineering, computer graphics, and any problem with quadratic relationships.

Inputs

Number

Root Index (2 = square, 3 = cube)

Results

Square Root

Perfect Root?

Yes

Result Squared

144

Last updated: May 29, 2026

Formula

**Square root:** √x is the value y such that y² = x. For positive x: y = √x means y ≥ 0 and y² = x. By convention, √x is always non-negative (the "principal" square root). **nth root:** ⁿ√x = x^(1/n) The value y such that y^n = x. **Worked examples:** √16 = 4 (because 4² = 16). √144 = 12 (because 12² = 144). ∛27 = 3 (because 3³ = 27). ⁴√16 = 2 (because 2⁴ = 16). √2 ≈ 1.41421... (irrational). **Perfect squares (memorize):** | n | n² | |---|---| | 1 | 1 | | 2 | 4 | | 3 | 9 | | 4 | 16 | | 5 | 25 | | 6 | 36 | | 7 | 49 | | 8 | 64 | | 9 | 81 | | 10 | 100 | | 11 | 121 | | 12 | 144 | | 13 | 169 | | 14 | 196 | | 15 | 225 | | 16 | 256 | | 20 | 400 | | 25 | 625 | **Common irrational square roots:** | x | √x | |---|---| | 2 | 1.41421... | | 3 | 1.73205... | | 5 | 2.23607... | | 6 | 2.44949... | | 7 | 2.64575... | | 10 | 3.16228... | | 50 | 7.07107... | | 100 | 10 (perfect square) | **Properties:** - √(a × b) = √a × √b. For example: √12 = √(4 × 3) = 2√3. - √(a/b) = √a / √b. For example: √(4/9) = 2/3. - √(x²) = |x| (absolute value for negative x). - (√x)² = x for x ≥ 0. - √x = x^(1/2). Exponent form. **Simplifying square roots:** For non-perfect squares, factor out perfect squares: √72 = √(36 × 2) = 6√2. √50 = √(25 × 2) = 5√2. √98 = √(49 × 2) = 7√2. **Worked example: simplify √200** 200 = 100 × 2. √200 = √100 × √2 = 10√2. Or factor: 200 = 4 × 50 = 4 × 25 × 2. √200 = 2 × 5 × √2 = 10√2. Same answer. **Common simplifications:** - √8 = 2√2 - √12 = 2√3 - √18 = 3√2 - √20 = 2√5 - √27 = 3√3 - √32 = 4√2 - √45 = 3√5 - √50 = 5√2 - √72 = 6√2 - √75 = 5√3 **Approximation methods:** For square roots of non-perfect squares, manual approximation: **Newton's method:** Start with guess y₀. Iterate: y_{n+1} = (y_n + x/y_n) / 2. For √50: Guess 7: (7 + 50/7)/2 = (7 + 7.143)/2 = 7.071. Next: (7.071 + 50/7.071)/2 = (7.071 + 7.071)/2 = 7.071. Converged: √50 ≈ 7.071. Very fast convergence — doubles correct digits each iteration. **Cube root:** ∛x is the value y such that y³ = x. Common cube roots: - ∛1 = 1 - ∛8 = 2 - ∛27 = 3 - ∛64 = 4 - ∛125 = 5 - ∛216 = 6 - ∛343 = 7 - ∛512 = 8 - ∛729 = 9 - ∛1000 = 10 Cube root accepts negative numbers (unlike square root): ∛(-8) = -2 (because (-2)³ = -8). **Fractional exponent form:** √x = x^(1/2). ∛x = x^(1/3). ⁿ√x = x^(1/n). For x^(m/n) = (x^m)^(1/n) = (x^(1/n))^m. Example: 8^(2/3) = (8^(1/3))² = 2² = 4. **Square roots in famous formulas:** - **Pythagorean theorem**: c = √(a² + b²). - **Distance formula**: d = √((x₂-x₁)² + (y₂-y₁)²). - **Quadratic formula**: x = (-b ± √(b² - 4ac))/(2a). - **Standard deviation**: σ = √(variance). - **Vector magnitude**: |v| = √(x² + y² + z²). - **RMS (root mean square)**: √(average of squares). **Imaginary unit:** √(-1) = i. Not a real number. Used in: - Complex numbers (a + bi). - AC circuit analysis (electrical engineering). - Quantum mechanics. - Signal processing. For negatives: √(-9) = √9 × √(-1) = 3i. **Approximate values to memorize:** - √2 ≈ 1.414 - √3 ≈ 1.732 - √5 ≈ 2.236 - √10 ≈ 3.162 Useful for mental math estimation. **Calculator buttons:** - **√**: square root. - **x²**: square (inverse of square root). - **^**: general exponent (use 1/n for nth root). - **y^x or x^y**: general power. **Educational notes:** Square roots typically introduced in: - 6th-8th grade: basic concept. - High school algebra: simplifying, equations. - High school geometry: Pythagorean theorem. - Calculus: derivatives of square root functions. **Common applications:** - **Geometry**: Pythagorean theorem, distance. - **Algebra**: quadratic formula, simplifying radicals. - **Statistics**: standard deviation. - **Physics**: kinetic energy (½mv²) inversions. - **Engineering**: stress and strain calculations. - **Computer graphics**: distance and length. - **Signal processing**: RMS values. **Square roots in real life:** - **TV diagonal**: from width and height: √(W² + H²). - **Distance on map**: √((Δx)² + (Δy)²). - **Investment doubling**: √(future_value/present_value) approximates per-period growth. - **Drinking glass capacity**: π × r² × h, with r derived from circumference. **Programming:** | Language | Square root | |---|---| | Python | math.sqrt(x) | | JavaScript | Math.sqrt(x) | | Java | Math.sqrt(x) | | C/C++ | sqrt(x) | | Excel | =SQRT(x) | | R | sqrt(x) | | MATLAB | sqrt(x) | For nth root: x^(1/n) in most languages. **Software:** - **Scientific calculators**: dedicated √ button. - **Spreadsheets**: SQRT function. - **Programming**: math libraries. - **Online**: instant for any number. **Pitfalls:** - **Negative input**: undefined for real numbers. - **Sign of result**: √4 = 2 (positive convention), but x² = 4 has solutions x = ±2. - **Simplifying**: extract perfect squares. - **Confusing √(a + b) with √a + √b**: not equal in general. - √(9+16) = √25 = 5, but √9 + √16 = 3 + 4 = 7. Not equal. - **For cube root of negative**: real result possible. - **For even roots**: even root of negative needs complex numbers. **Historical notes:** - Babylonians knew √2 (~1900 BC). - Pythagoreans proved √2 irrational (~500 BC) — shocking discovery. - Newton's method (1669): efficient computer approximation. - Modern: high-precision libraries compute to millions of digits if needed. **Pitfalls (continued):** - **For negative number**: no real square root. - **Calculator domain errors**: trying √(-1) in calculator gives error or "i". - **Distinguishing rational vs irrational**: many roots are irrational. - **Approximations vs exact**: decimal display may hide irrational nature. - **For very large numbers**: precision limits in floating-point.

How to use this calculator

Enter a non-negative number to find square root.
Specify root index (2 for square, 3 for cube, etc.).
Calculator returns the root.
For irrational roots: decimal approximation given.
Verify: result raised to root index should give original number.
For negative numbers: only odd roots are real.

Worked examples

Standard square root

**Scenario:** Find √144. **Calculation:** Look for n where n² = 144. n = 12 (since 12 × 12 = 144). **Result:** √144 = 12. Perfect square — exact integer result. Verify: 12 × 12 = 144 ✓.

Irrational square root

**Scenario:** Find √50. **Calculation:** 50 is not a perfect square. Simplify: √50 = √(25 × 2) = 5√2. Decimal: 5 × 1.41421... ≈ 7.0711. **Result:** √50 = 5√2 ≈ 7.0711. Exact form 5√2; decimal approximate. Verify: 7.0711² ≈ 50.001 ✓ (small rounding).

Cube root

**Scenario:** Find ∛216. **Calculation:** Look for n where n³ = 216. n = 6 (6 × 6 × 6 = 216). **Result:** ∛216 = 6. Perfect cube. Verify: 6³ = 216 ✓. For non-perfect cubes: use approximation methods.

When to use this calculator

**Use square root for:**

- **Pythagorean theorem**: c = √(a² + b²). - **Distance formula**: distance from two points. - **Quadratic formula**: ± √(b² - 4ac). - **Standard deviation**: square root of variance. - **Vector magnitude**: length of vector. - **Geometry**: side lengths from areas. - **Physics**: deriving velocity from kinetic energy. - **Engineering**: many formulas involve roots.

**Recognizing perfect squares:**

Numbers expressible as n × n: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225...

Mental recognition speeds up square root calculations.

**Simplifying surds:**

For non-perfect squares, extract any perfect square factors: - √12 = 2√3. - √48 = 4√3. - √200 = 10√2.

Always simplify to keep exact form.

**Common operations:**

- **Multiplication**: √a × √b = √(ab). - **Division**: √a / √b = √(a/b). - **Squaring**: (√x)² = x (for x ≥ 0). - **Addition**: √a + √b ≠ √(a+b) — distinct values.

**Common applications:**

- **Math homework**: Pythagorean theorem, simplifying. - **Statistics**: standard deviation, RMS. - **Physics**: deriving variables from quadratic equations. - **Engineering**: stress, strain, frequency calculations. - **Construction**: roof rafters (Pythagorean). - **Computer graphics**: distance calculations. - **Finance**: not commonly, but appears in option pricing.

**Approximation tips:**

For estimating √n: - Find nearest perfect squares. - Estimate between them.

For √70: between √64 = 8 and √81 = 9. Closer to 8: ~8.4. Actual: 8.367.

For more precision: use Newton's method or calculator.

**Imaginary numbers:**

When dealing with √ of negative: √(-9) = 3i. Used in: AC circuits, quantum mechanics, signal processing.

For purely real-valued problems, ensure inputs are non-negative.

**Software:**

- **Calculators**: SQRT button. - **Python**: math.sqrt or x**0.5. - **Excel**: =SQRT(A1). - **Programming**: any language has square root function. - **Wolfram Alpha**: also gives exact form (e.g., √50 = 5√2).

**Pitfalls:**

- **Square root of sum ≠ sum of roots**: √(9+16) = 5, not 3+4 = 7. - **Negative numbers**: no real square root. - **Sign convention**: √4 = 2 (positive only); x² = 4 → x = ±2 (both). - **Confusing roots and exponents**: √x = x^(1/2). - **For cube root of negative**: real (∛(-8) = -2). - **For even roots of negative**: complex (√(-4) = 2i). - **Mixing exact and decimal**: 5√2 (exact) vs 7.071 (approximate).

**Practical examples:**

- **TV size**: 32" TV at 16:9 ratio: diagonal 32, side = 32 × 16/√(16²+9²) ≈ 27.9". - **Standard deviation**: variance σ²; standard deviation σ = √σ². Most statistics tables show σ. - **Pythagorean shortcuts**: 3-4-5 triangle: legs 3, 4; hypotenuse = √25 = 5. - **Period of pendulum**: T = 2π√(L/g).

**Educational notes:**

Square roots foundational for: - Pythagorean theorem (high school geometry). - Quadratic formula (algebra). - Distance calculations (coordinate geometry). - Standard deviation (statistics). - Many calculus and physics formulas.

Master square root operations early; foundation for higher math.

**Pitfalls (continued):**

- **For very small numbers**: square root reduces precision. - **For very large numbers**: result may be huge. - **Calculator notation**: shows decimal, but real value may be irrational. - **For algebraic manipulation**: keep exact form when possible.

Common mistakes to avoid

Computing √ of negative number expecting real result.
Treating √(a+b) as √a + √b (they're NOT equal).
Forgetting that x² = 4 has two solutions: x = ±2.
Mixing simplification: √8 → 2√2, not 4√2.
Computing exponent and root in wrong order: x^(2/3) = (x^2)^(1/3) = (x^(1/3))^2.
For cube root: applying square root rules.
For very large numbers: losing precision in approximations.
Confusing nth root with division by n.

Frequently Asked Questions

Sources & further reading

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