Square Root Calculator

About Square Root Calculator

How to Simplify a Square Root

Simplifying a radical means pulling perfect square factors out from under the root sign. The process works like this:

1. Factor the number into a product that includes the largest perfect square
2. Take the square root of the perfect square factor
3. Write it outside the radical, leaving the remainder inside

Worked example: Simplify √72

1. Find the largest perfect square factor of 72: 36 × 2 = 72
2. √72 = √(36 × 2) = √36 × √2
3. √36 = 6, so √72 = 6√2
4. As a decimal: 6 × 1.41421... = 8.48528...

More examples:

√n	Factorisation	Simplified Form	Decimal
√50	25 × 2	5√2	7.07107
√48	16 × 3	4√3	6.92820
√200	100 × 2	10√2	14.14214
√98	49 × 2	7√2	9.89949
√75	25 × 3	5√3	8.66025
√180	36 × 5	6√5	13.41641
√12	4 × 3	2√3	3.46410

Perfect Squares Reference

A perfect square has an integer square root. Memorising the first 20 is useful for quick mental maths and simplifying radicals:

n	n²	n	n²	n	n²	n	n²
1	1	6	36	11	121	16	256
2	4	7	49	12	144	17	289
3	9	8	64	13	169	18	324
4	16	9	81	14	196	19	361
5	25	10	100	15	225	20	400

Rational vs Irrational Roots

A root is rational if it can be written as a fraction (whole numbers and terminating decimals). It is irrational if the decimal goes on forever without repeating.

Type	Example	Why
Rational	√25 = 5	25 is a perfect square
Rational	√(9/16) = 3/4	Both numerator and denominator are perfect squares
Irrational	√2 = 1.41421356...	2 is not a perfect square
Irrational	√3 = 1.73205080...	3 is not a perfect square
Irrational	√7 = 2.64575131...	7 is not a perfect square

The proof that √2 is irrational was one of the earliest results in mathematics, attributed to Hippasus around 500 BC. In fact, the square root of any non-perfect-square integer is irrational.

Cube Roots and How They Work

A cube root asks: what number multiplied by itself three times gives this value?

∛x = y means y × y × y = x

Worked example: Simplify ∛54

1. Find the largest perfect cube factor: 27 × 2 = 54
2. ∛54 = ∛(27 × 2) = ∛27 × ∛2
3. ∛27 = 3, so ∛54 = 3∛2
4. As a decimal: 3 × 1.25992... = 3.77976...

Perfect cubes to know:

n	n³	n	n³
1	1	6	216
2	8	7	343
3	27	8	512
4	64	9	729
5	125	10	1,000

Unlike square roots, cube roots of negative numbers are real. ∛(-8) = -2 because (-2) × (-2) × (-2) = -8.

Nth Roots

The nth root generalises the concept: the nth root of x is the number that, raised to the power n, gives x.

Root	Notation	As Exponent	Example
Square root	√x	x^(1/2)	√16 = 4
Cube root	∛x	x^(1/3)	∛27 = 3
Fourth root	⁴√x	x^(1/4)	⁴√81 = 3
Fifth root	⁵√x	x^(1/5)	⁵√32 = 2
Sixth root	⁶√x	x^(1/6)	⁶√64 = 2

Even roots (2nd, 4th, 6th) of negative numbers are not real. Odd roots (3rd, 5th, 7th) of negative numbers are real and negative.

Where Square Roots Appear

Application	Formula	Why Roots Are Needed
Pythagorean theorem	c = √(a² + b²)	Finding the hypotenuse or side of a right triangle
Distance formula	d = √((x₂-x₁)² + (y₂-y₁)²)	Distance between two points on a plane
Quadratic formula	x = (-b ± √(b²-4ac)) / 2a	The discriminant is under a square root
Standard deviation	σ = √(variance)	Variance is in squared units, root converts back
RMS voltage	V_rms = V_peak / √2	Root mean square for AC circuits
Escape velocity	v = √(2GM/r)	Speed needed to leave a gravitational field

Roots and the Exponent Connection

Every root can be written as a fractional exponent, which is the key to handling roots in algebra:

ⁿ√x = x^(1/n)

This means all exponent rules apply to roots. For example, √a × √b = √(ab), and √(a/b) = √a / √b. The exponent calculator handles any base raised to any power, including fractional exponents. For problems involving right triangles, the Pythagorean theorem calculator uses square roots to find missing sides, and the quadratic calculator applies the discriminant rule b² - 4ac under a radical.

How the Calculator Finds Simplified Radicals

The algorithm factors the number under the radical and pulls out perfect-square factors one by one. It trial-divides from 2 upward, checking whether the remaining inside value is divisible by f². Each time it is, f moves outside the radical and f² divides out from inside. The loop stops at √n because any prime factor above √n can appear at most once, so it cannot form a perfect square.

Trace for √200:

1. Start: outside = 1, inside = 200
2. f = 2: 200 / 4 = 50, so outside = 2, inside = 50. Check again: 50 / 4 = 12.5 (not divisible), stop for f = 2.
3. f = 3: 50 / 9 is not an integer, skip.
4. f = 4: already handled via f = 2 (since 4 = 2²).
5. f = 5: 50 / 25 = 2, so outside = 2 × 5 = 10, inside = 2. Check again: 2 / 25 is less than 1, stop.
6. f² = 36 > 2, loop ends.
7. Result: 10√2, which matches 14.14214 as a decimal.

This is the same method used in CAS engines like SymPy's sqrtdenest and Mathematica's RadicalDenest, though those also handle nested radicals such as √(5 + 2√6) = √2 + √3.

Common Mistakes When Simplifying Roots

The three mistakes that catch most students out are:

• Stopping at the first perfect square factor. For √72, some students factor as 4 × 18 and write 2√18. That is technically correct but not fully simplified because 18 still contains 9 as a factor. Always pull out the LARGEST perfect square, or repeat until nothing factors.
• Splitting a sum under a radical. √(a + b) is NOT √a + √b. For example, √(9 + 16) = √25 = 5, not 3 + 4 = 7. Roots distribute over multiplication and division, never over addition or subtraction.
• Forgetting the ± sign when solving x² = k. The principal square root is always positive, but the equation x² = 25 has two solutions: x = 5 and x = -5. The square root symbol by itself returns only the positive root. This is why the quadratic formula writes ±√(b² - 4ac).

A fourth subtle error: √(x²) equals |x|, not x. If x is negative, squaring removes the sign and the root cannot recover it. Most calculators, including this one, assume non-negative inputs and return the principal root.

Square Roots in History and Culture

The Babylonian clay tablet YBC 7289 (c. 1800-1600 BC) shows √2 computed to five decimal places as 1.41421, an accuracy that was not improved on in Europe for 3,000 years. According to the Yale Babylonian Collection, the tablet uses a numerical method equivalent to Newton's iteration: xₙ₊₁ = (xₙ + n/xₙ)/2. Starting from x₀ = 1.5 and iterating gives 1.41666... then 1.41421... in just two steps.

The Pythagoreans in Ancient Greece proved √2 was irrational around 500 BC. The proof by contradiction - assume √2 = p/q in lowest terms, show both p and q must be even, contradiction - is still taught today in UK A-level Further Maths and US college algebra courses. The Pythagorean Hippasus is traditionally credited (and, per legend, drowned for revealing the result). Modern number theory, per the Britannica entry on irrational numbers, confirms that the square root of any non-perfect-square integer is irrational, extending the Pythagorean result to all such roots.

Newton's method (sometimes called the Newton-Raphson method) is the algorithm your pocket calculator and JavaScript's Math.sqrt use internally, typically in hardware via the x86 SQRTSD instruction or its ARM equivalent. The IEEE 754 standard guarantees Math.sqrt returns a correctly-rounded result within 0.5 ULP (unit in the last place).

Square Roots in Everyday Engineering

Square roots show up everywhere in applied maths, often hidden inside formulas people use without realising. A few concrete examples:

• Screen sizing. A "55-inch" TV measures its diagonal. For a 16:9 panel, width = 55 × 16 / √(16² + 9²) ≈ 47.9 inches and height ≈ 27.0 inches. Without the square root, you can't convert diagonal size to physical area.
• GPS accuracy. The "horizontal dilution of precision" reported by GPS receivers uses square roots of covariance matrix elements. A DOP of 1.0 means 1-metre satellite error gives 1-metre position error; higher values multiply the uncertainty.
• Photography depth of field. Hyperfocal distance is proportional to √(aperture × focal length²), which is why doubling aperture f-number quadruples depth of field at a given focus distance.
• Finance volatility. Annualising standard deviation from daily returns uses √252 (trading days per year) as the scaling factor, per the CFA Institute's Level I curriculum.
• Water flow. The orifice flow equation Q = Cd × A × √(2gh) uses a square root to relate discharge to head pressure, per the USGS Water Supply Paper 2175.

Any time a formula contains a squared term on one side and a linear term on the other, rearranging it will produce a square root. This is why √ appears in relativity (Lorentz factor), quantum mechanics (wavefunction normalisation), and statistics (the Z-score involves a denominator of √n).

Sources

• Wolfram MathWorld - Square Root
• Britannica - Irrational Number
• Yale Babylonian Collection - YBC 7289 (Square Root of 2)
• MDN - Math.sqrt (IEEE 754 behaviour)
• Wikipedia - Methods of Computing Square Roots
• SymPy - sqrtdenest Documentation

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