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.

Last updated: May 31, 2026

About this calculator

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.

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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Convert numbers to and from scientific notation.

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Count significant figures in a number and round to a specific number of sig figs.

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About this calculator

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.

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

Formula

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

Cube root

When to use this calculator

**Use square root for:**

**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:**

**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:**

**Pitfalls:**

**Practical examples:**

**Educational notes:**

Master square root operations early; foundation for higher math.

**Pitfalls (continued):**

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

About this calculator

Formula

How to use this calculator

Worked examples

Standard square root

Irrational square root

Cube root

When to use this calculator

Common mistakes to avoid

Frequently Asked Questions

What is a square root?

Can you take the square root of a negative number?

How do I simplify √50?

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

How do I find a square root without a calculator?

How are square roots used in geometry?

What's the square root of 2?

How are radicals different from exponents?

Sources & further reading

Related Calculators

Exponent Calculator

Pythagorean Theorem Calculator

Logarithm Calculator

Scientific Notation Calculator

Significant Figures Calculator

Quadratic Formula Calculator

Inputs

Results

About this calculator

Formula

How to use this calculator

Worked examples

Standard square root

Irrational square root

Cube root

When to use this calculator

Common mistakes to avoid

Frequently Asked Questions

What is a square root?

Can you take the square root of a negative number?

How do I simplify √50?

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

How do I find a square root without a calculator?

How are square roots used in geometry?

What's the square root of 2?

How are radicals different from exponents?

Sources & further reading

Related Calculators

Exponent Calculator

Pythagorean Theorem Calculator

Logarithm Calculator

Scientific Notation Calculator

Significant Figures Calculator

Quadratic Formula Calculator

Inputs

Results