Square Root Calculator

Find square roots, cube roots, and any nth root instantly — plus the simplified radical form (like √72 = 6√2), a perfect-square check, and the square and cube of your number for context. Roots that are not whole numbers are shown to full precision, never rounded away silently.

What Is a Square Root?

The square root of a number is the value that, multiplied by itself, gives the original number. √25 = 5 because 5 × 5 = 25. Every positive number has two square roots — a positive and a negative one (both 5 and −5 square to 25) — but calculators and textbooks conventionally return the positive one, called the principal root. Cube roots undo cubing (∛27 = 3 because 3 × 3 × 3 = 27), and the nth root generalises the idea: the 4th root of 81 is 3, because 3⁴ = 81.

Roots that are whole numbers are tidy, but most are not. √2 = 1.41421356… never terminates and never repeats — it is irrational, a fact the ancient Greeks found so disturbing that legend says it was kept secret. Your calculator handles both cases: exact whole answers when they exist, and full-precision decimals otherwise. The Khan Academy root lessons explain the geometry behind squares and roots, and Wolfram MathWorld documents nth roots and radicals formally.

Simplified Radical Form

√72 is not a whole number, but it can be written exactly as 6√2. The method: factor the number, pull out squared factors in pairs, and leave the rest under the radical. 72 = 36 × 2 = 6² × 2, so √72 = √(36 × 2) = 6√2. Simplified form matters in algebra, where exact answers beat rounded decimals — solving x² = 72 gives x = ±6√2, and a decimal would lose exactness. The calculator above shows the simplified form for square roots of whole numbers automatically.

Squares and Cubes, 1–25

Memorising the squares up to 25 pays off constantly: it makes perfect-square checks instant and estimation far faster. A number between 144 and 169 has a root between 12 and 13, for example — a useful sanity check on any root calculation.

Tips for Working With Roots

• Estimate first. √90 sits between √81 (9) and √100 (10), closer to 9.5 — if your calculator says 3, something is wrong.
• Keep exact form in algebra. 6√2 is exact; 8.4853 is rounded. Use the radical form until the final step.
• Even roots of negatives are imaginary. √−9 = 3i, where i² = −1. Odd roots of negatives stay real: ∛−27 = −3.
• Roots undo powers, not multiplication. √(a × b) = √a × √b works, but √(a + b) does not equal √a + √b.

Related tools: the scientific calculator for roots inside bigger expressions, the fraction calculator for exact fractional work, and the average calculator for statistics.

Estimating Roots by Hand: Bracket and Refine

You can estimate any square root with nothing but squaring and patience. Take √90. First bracket it: 9² = 81 and 10² = 100, so the answer sits between 9 and 10, closer to 9.5. Try 9.5: 9.5² = 90.25, slightly too high, so the root is slightly below 9.5. Try 9.48: 9.48² = 89.8704, slightly too low. Try 9.49: 9.49² = 90.0601, slightly too high. The root is between 9.48 and 9.49, about 9.487. Each step halves the uncertainty, and two or three refinements beat most people’s mental precision needs.

The same method works for cube roots, bracketing between cubes instead of squares: ∛50 sits between 3³ = 27 and 4³ = 64, closer to 4, and refining gives about 3.68 (3.68³ = 49.84, close to 50). The point of doing this by hand once is not to replace the calculator — it is to own a sanity check. If a tool ever returns √90 = 3, your bracket instantly says impossible, because 3² = 9, not 90. Estimation is the cheapest error detection there is.

Roots in Geometry, Building and Science

Roots appear wherever squares and areas appear. A square garden with an area of 144 square metres has sides of √144 = 12 metres, and fencing it needs 4 × 12 = 48 metres of fence. A right-angled triangle with sides 6 and 8 has a hypotenuse of √(36 + 64) = √100 = 10, the classic 6-8-10 triangle that builders use to check square corners on site: measure 6 units along one wall, 8 along the other, and the diagonal must be 10, or the corner is not square.

Science uses roots the same way. The time for an object to fall a height h is √(2h ÷ g), where g is 9.8 metres per second squared. Drop something from 20 metres and the fall takes √(40 ÷ 9.8) = √4.08 ≈ 2.02 seconds. The square root appears because distance under constant acceleration grows with the square of time — undoing a square always means taking a root. Whenever a formula squares something, expect a root when you solve it backwards, and the scientific calculator handles roots inside longer expressions.

Cube Roots and Nth Roots in Everyday Maths

Cube roots answer “how big a box holds this volume?” A box holding 125 litres has sides of ∛125 = 5 units, and doubling a cube’s volume does not double its side: the new side is the old side × ∛2, about 1.26 times. That 1.26 factor surprises people, and it is the same reason doubling a pizza’s diameter quadruples its area — lengths, areas, and volumes scale differently, and roots are the conversion between them.

Higher roots show up in growth and finance. If an investment grows to 16 times its starting value over 4 equal years, each year’s growth factor is the 4th root of 16, which is 2 — the money doubled every year. The nth root is simply the undo button for “to the power n”, and this calculator’s nth-root mode covers any degree from 2 to 100, including the negative-number rule: odd roots of negatives stay real, even roots do not.

Scaling works the same way in reverse. Double the side of a square and its area becomes four times larger, because 2² = 4; double the side of a cube and its volume becomes eight times larger, because 2³ = 8. This is why a 12-inch pizza is not twice as much pizza as a 6-inch one — it is four times as much — and why a slightly larger pot holds a surprisingly larger meal. Roots undo exactly this: given the area, the root gives the side; given the volume, the cube root gives the edge.

Perfect Squares Beyond 25

Knowing the squares up to 25 is a good start, but roots of larger numbers become far easier once you know the squares from 26 to 40 as well. A number such as 1,024 is instantly recognisable as 32 squared, and 1,225 is 35 squared. That recognition turns a slow calculation into a one-step answer, and it gives you brackets for every number in between: any root between 900 and 1,024 must lie between 30 and 32, and any root between 1,225 and 1,296 must lie between 35 and 36.

Use the table to bracket a root before you reach for a calculator. The number 1,000 sits between 961 and 1,024, so its root sits between 31 and 32, and it is closer to 32: the gap from 961 up to 1,000 is 39, while the gap from 1,000 up to 1,024 is only 24. The calculator gives about 31.62, which fits. Likewise 1,200 sits between 34 squared, 1,156, and 35 squared, 1,225, so its root is about 34.64. Bracketing never gives the final digits, but it always tells you whether a calculator answer is in the right neighbourhood.

• Instant recognition. Spot 729, 841, 1,089, or 1,444 in a problem and you know the root at once: 27, 29, 33, and 38.
• Faster bracketing. Two nearby squares pin any root between two whole numbers, which is the first step of every hand-estimation method on this page.
• Safer checking. If a result for the root of 1,300 comes out below 36, the table says it must be wrong, because 36 squared is only 1,296.

How to Simplify a Radical, Step by Step

Simplifying a radical means pulling every squared factor out of the root so that the number left under the radical has no squared factor remaining. The result is exact, unlike a rounded decimal, and it is the form algebra expects. The calculator does this automatically for square roots of whole numbers, but the steps are worth knowing because they explain why the answer looks the way it does.

1. Factor the number. Break it into small factors. For 200, write 200 as 100 times 2, or as 2 times 2 times 2 times 5 times 5.
2. Group squared factors in pairs. Each pair of equal factors is one squared factor. In 200, the pair of 2s and the pair of 5s give 4 times 25, which is 100, a perfect square.
3. Move one factor from each pair outside the root. The square root of 100 is 10, so 10 comes out, and the unpaired 2 stays inside: 10 times the square root of 2.
4. Check by squaring. Square the answer and confirm you get the original number. Ten times the square root of 2, squared, is 100 times 2, which is 200.

The same steps simplify the examples in the table below. Decimals in the table are rounded to four decimal places for reading; the calculator above shows results to 8 decimal places, and the radical form itself is exact at any precision.

Simplifying Radicals With More Than One Squared Factor

Sometimes the squared factor is not obvious at first glance. Take the square root of 360. Factoring gives 360 as 36 times 10, and 36 is 6 squared, so the root is 6 times the square root of 10, about 18.9737. Take the square root of 288: 288 is 144 times 2, and 144 is 12 squared, so the root is 12 times the square root of 2, about 16.9706. In both cases the check is the same: square the outside number, multiply by the number left inside, and you must land back on the original number, since 36 times 10 is 360 and 144 times 2 is 288. If a number has no squared factor at all, apart from 1, it cannot be simplified: the square root of 30, the square root of 42, and the square root of 70 stay exactly as they are.

Cube Roots and Nth Roots Explained

A cube root undoes cubing, just as a square root undoes squaring. The cube root of 27 is 3, the cube root of 64 is 4, and the cube root of 1,000 is 10. The nth root generalises the same idea to any power: the 4th root of a number is the value that gives the number when multiplied by itself four times, the 5th root is the value that gives the number when multiplied by itself five times, and so on. Written with powers, the nth root of x is x raised to the power 1 divided by n, which is why the scientific calculator can find roots with its power key as well.

How Nth Roots Work

The defining test never changes: raise the answer to the stated power and you must get the original number back. The 4th root of 81 is 3 because 3 times 3 times 3 times 3 is 81. The 5th root of 32 is 2 because 2 raised to the power 5 is 32. The 6th root of 64 is 2 because 2 raised to the power 6 is 64. Higher roots grow slowly: the 10th root of 1,024 is 2, because 2 raised to the power 10 is 1,024, so even a four-digit number can have a small nth root when n is large. The sign rule from the earlier sections applies at every degree: odd roots of negative numbers are negative and real, while even roots of negative numbers are not real numbers at all.

Cube Roots of Fractions and Decimals

Cube roots follow the same fraction rule as square roots: root the top and root the bottom separately. The cube root of 1 divided by 8 is 1 divided by 2, because 1 divided by 2, cubed, is 1 divided by 8. The cube root of 0.125 is 0.5, as the table shows. For growth problems the nth root is the natural tool: if a quantity grows to 16 times its starting value over 4 equal periods, each period multiplies it by the 4th root of 16, which is 2, so the quantity doubles each period. The compound interest calculator handles the money version of that growth, while this page handles the root itself.

Square Roots of Decimals and Fractions

Roots of decimals and fractions obey one simple rule: for a fraction, take the root of the numerator and the root of the denominator separately. The square root of 9 divided by 16 is 3 divided by 4, because 3 divided by 4, squared, is 9 divided by 16. The square root of 25 divided by 36 is 5 divided by 6. Decimals work the same way once you count places: the square root of 0.09 is 0.3, because 0.3 times 0.3 is 0.09, and the square root of 1.44 is 1.2, because 1.2 times 1.2 is 1.44.

The surprising case is a decimal below 1, whose root is larger than the decimal itself. The square root of 0.25 is 0.5, which is larger than 0.25. That feels backwards until you remember that squaring a number below 1 makes it smaller: 0.5 squared is 0.25, so undoing the square must go back up to 0.5. The same happens with fractions: the square root of 1 divided by 4 is 1 divided by 2, which is larger than 1 divided by 4. Roots shrink numbers above 1 and grow numbers between 0 and 1, and 1 itself is the fixed point where the root equals the number.

1. Split a fraction. Root the numerator and the denominator separately, and simplify each part on its own.
2. Count decimal places. A root of a decimal with two places has one place: the root of 0.09 has one decimal place, 0.3, because squaring doubles the places.
3. Keep the radical when it will not simplify. If the numerator or denominator is not a perfect square, leave the radical in the answer, as with the square root of 2 divided by 9, which is the square root of 2 over 3.

For exact fractional arithmetic around a root problem, the fraction calculator keeps numerators and denominators exact instead of rounding them into decimals part-way through.

Irrational Numbers: Why Most Roots Never End

Most square roots are irrational numbers: their decimals never terminate and never settle into a repeating pattern. The square root of 2 is 1.41421356 and continues without end, and no fraction of two whole numbers equals it exactly. The classic argument is short. Suppose the square root of 2 equalled a fraction in lowest terms, p divided by q, with no common factor. Then p squared would be twice q squared, so p squared would be even, which forces p itself to be even. Writing p as twice some whole number then forces q to be even as well, contradicting the claim that the fraction was in lowest terms. No such fraction exists, so the square root of 2 cannot be written as a fraction at all.

That argument only fails when the number is a perfect square. The square root of 4 is 2, the square root of 9 is 3, and the square root of 144 is 12, all whole numbers and therefore rational. Every other whole number has an irrational square root. The same holds for cube roots and higher roots: the cube root of 8 is 2, but the cube root of 9 is irrational. Irrational does not mean vague or approximate: the square root of 2 is a perfectly definite number, the exact length of the diagonal of a square with side 1, and the radical sign is its exact name. Only its decimal expansion is endless.

• Perfect squares are the exception. If the number is a perfect square, its root is a whole number. If it is not, the root is irrational.
• A terminating decimal is a warning sign. A calculator display always stops somewhere, so a display that stops does not prove the number stops. The calculator above shows 8 decimal places; for an irrational root, every one of those places is still an approximation.
• Keep the radical for exact work. In algebra and geometry, the square root of 2 is the exact answer, and 1.4142 is only an approximation of it. Use the radical until the last step, then round once.

Estimating Square Roots by Hand: The Averaging Method

Bracketing narrows a root down, but the averaging method, known for thousands of years and sometimes called the Babylonian method, converges far faster. Start with any reasonable guess, divide the number by the guess, and average the guess with the result. Repeat, and each round roughly doubles the number of correct digits. The logic is simple: if your guess is too low, the division result is too high by a matching amount, so the average sits between them and closer to the truth than either one.

Find the square root of 50. Start with a guess of 7, since 7 squared is 49. Divide 50 by 7 to get about 7.1429. Average 7 and 7.1429 to get about 7.0714. Square 7.0714 and you get about 50.005, already within 0.005 of 50 after a single round. Repeat once: divide 50 by 7.0714 to get about 7.0707, average the two to get about 7.0711, and that matches the true value, 7.0710678, to four decimal places. Two rounds of school-level arithmetic reached calculator quality.

1. Guess. Pick a whole number whose square is near the target. For 50, guess 7.
2. Divide. Divide the target by the guess. For 50 divided by 7, the result is about 7.1429.
3. Average. Average the guess and the division result. The average, about 7.0714, is the next guess.
4. Repeat or stop. Square the new guess to check it. If it is close enough for the job, stop; if not, divide and average again.

One more short example shows the speed. Estimate the square root of 20, starting from 4.5, since 4.5 squared is 20.25. Divide 20 by 4.5 to get about 4.4444, average 4.5 and 4.4444 to get 4.4722, and that is already within 0.0001 of the true value, 4.4721. A single averaging step turned a rough guess into a four-decimal answer.

• Checking a calculator result. One averaging round from a rough guess confirms whether a displayed answer is plausible.
• Exams without a calculator. Two rounds give more accuracy than most exam questions require.
• Rough measures on site. When a length must be estimated from an area, one round is usually more precision than a tape measure can use.

Pythagoras and Area: Roots in Real Shapes

Pythagoras gives the most famous use of square roots: in a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. Finding the hypotenuse means adding squares and rooting the total, but finding a missing shorter side means subtracting squares and rooting the difference. If the hypotenuse is 13 and one side is 5, the other side is the square root of 169 minus 25, which is the square root of 144, which is 12. That 5-12-13 triangle is the second of the classic triples every builder and student should recognise, and the table below collects the most useful ones.

Finding a Missing Side

Work backwards with the same rule. A ladder leaning against a wall forms a right-angled triangle: if the ladder is 10 long and its foot is 6 from the wall, the height it reaches is the square root of 100 minus 36, which is the square root of 64, which is 8. A rectangle 9 wide and 12 tall has a diagonal of the square root of 81 plus 144, which is the square root of 225, which is 15. Whenever two sides of a right-angled triangle are known, the third is one square root away, and the calculator above finds it directly by rooting the sum or the difference of the two squares.

From Area Back to Side Length

Area problems run the root in the other direction. A square plot with an area of 225 square metres has sides of 15 metres, because 15 squared is 225, and fencing it needs 4 times 15, or 60 metres of fence. A square room of 625 square feet has 25-foot sides. For a circle, the radius is the square root of the area divided by pi: a circle with an area of about 314.16 has a radius of about 10, because 314.16 divided by pi is about 100, and the square root of 100 is 10. When a shape is scaled up, lengths, areas, and volumes scale by different factors, and the ratio calculator keeps the proportions straight while the root converts an area back into a length.

Common Mistakes With Square Roots

Roots cause a small set of errors again and again, and each one has a one-line correction. Learning the corrections is cheaper than reworking every problem they spoil.

• Distributing a root over addition. The square root of 9 plus 16 is the square root of 25, which is 5, not the square root of 9 plus the square root of 16, which is 3 plus 4, or 7. Add first, then root.
• Adding unlike roots. The square root of 3 plus the square root of 3 is 2 times the square root of 3, about 3.464, not the square root of 6, about 2.449. Only like radicals combine, like like terms.
• Confusing a root with halving. The square root of 16 is 4, not 8. Halving divides by 2; rooting undoes squaring.
• Forgetting the negative solution of an equation. The equation x squared equals 49 has two solutions, 7 and negative 7, because both square to 49. The root symbol alone still means the positive root, so equations must write plus-or-minus explicitly.
• Dropping the absolute value. The square root of x squared is the absolute value of x, not x itself: the square root of negative 5 squared is the square root of 25, which is 5, not negative 5.
• Rooting a decimal the wrong way. The square root of 0.25 is 0.5, not 0.05, because 0.05 squared is only 0.0025. Count the decimal places: squaring doubles them, so rooting halves them.
• Taking an even root of a negative and calling it negative. The square root of negative 9 is not negative 3, because negative 3 squared is positive 9. Even roots of negatives are imaginary, not negative.
• Rounding too early. Rounding the square root of 2 to 1.41 and then squaring gives 1.9881, not 2. Keep the radical or the full precision until the final step.

Rules for Multiplying, Dividing and Adding Roots

Roots combine under a few clean rules, and each rule is just the power rule in disguise. The product rule says the root of a product is the product of the roots, which is what makes simplifying radicals possible in the first place: the square root of 72 is the square root of 36 times the square root of 2, which is 6 times the square root of 2. The quotient rule does the same for division, and the addition rule is the strict one: roots add only when the radical part matches.

• Product: √a × √b = √(a × b). Example: √3 × √12 = √36 = 6.
• Quotient: √a ÷ √b = √(a ÷ b). Example: √18 ÷ √2 = √9 = 3.
• Like radicals add: a√c + b√c = (a + b)√c. Example: 2√3 + 5√3 = 7√3.
• Squaring undoes rooting: (√a)² = a. Example: (√7)² = 7, and (6√2)² = 36 × 2 = 72.
• Pulling out a square: √(a² × b) = a√b. Example: √(25 × 2) = 5√2.
• No distribution over addition: √(a + b) is not √a + √b, and √(a − b) is not √a − √b.

These rules also explain why the simplified form is unique: once every squared factor has been pulled out, no further rule can change the radical part, so two people simplifying the same root must arrive at the same answer. If two answers look different, square both and compare, or simplify both fully and compare the radical parts.

Real-Life Uses of Square Roots

Roots rarely announce themselves in everyday life, but they sit behind a surprising range of ordinary jobs. Any question that starts from a squared quantity and asks for the original length, rate, or size is a root question, even when nobody writes a radical sign.

• Building and DIY. Checking a corner is square with the 6-8-10 triangle, finding a diagonal brace length, and converting a floor area into a side length all use the root, as the Pythagoras section shows.
• Tiling and flooring. A room of 144 square feet tiled with a square layout has 12-foot sides, which sets out the tile grid before a single tile is cut.
• Finance and growth. The average yearly growth rate over several years is an nth root of the total growth factor, and the percentage calculator handles the percentage side of the same problems.
• Statistics. The standard deviation, the most common measure of spread, is a square root: it is the root of the average squared distance from the mean, which is why it is measured in the same units as the data. The average calculator finds the mean that the calculation starts from.
• Physics. Fall time, pendulum period, and escape speed all contain roots, because each undoes a squaring in the underlying law.
• Photography. Lens aperture numbers run 1, 1.4, 2, 2.8, 4, and so on, each step multiplying by the square root of 2, about 1.414, because each full stop must double the area of the opening, and area grows with the square of the diameter.
• Health and fitness. Body mass index divides weight by height squared, so solving the formula backwards for a target weight or height involves a root. The BMI calculator does the forward calculation.
• Distance on a map or screen. The straight-line distance between two points is the square root of the sum of the squared horizontal and vertical differences, which is Pythagoras again, applied to coordinates.

How to Check Any Root Answer

1. Square it back. The answer, squared, must return the original number, allowing for the rounding you used.
2. Bracket it. The answer must sit between the roots of the two perfect squares nearest the original number.
3. Estimate it. One round of the averaging method should land near the displayed answer; a large gap means one of the two is wrong.
4. Compare the simplified form. If the answer was given as a radical, square the outside number, multiply by the number inside, and confirm the product is the original number.

Frequently Asked Questions

Why do calculators only show the positive root?

By convention, the square root symbol means the principal root, which is the positive one. The equation x² = 25 genuinely has two solutions, +5 and −5, because both square to 25. But a function must return one value, so calculators return +5, and algebra problems that need both answers write ±5 explicitly.

What is the square root of 0 and of 1?

√0 = 0, because 0 × 0 = 0, and √1 = 1, because 1 × 1 = 1. They are the only two whole numbers that are their own square roots, and they anchor the scale: every positive number below 1 has a root larger than itself (√0.25 = 0.5), while every number above 1 has a root smaller than itself.

What is the square root of a negative number?

Not a real number. Even roots of negatives are imaginary: √−9 = 3i. Odd roots of negatives are real: ∛−27 = −3.

How do I simplify √72?

Factor it: 72 = 36 × 2, and 36 is 6², so √72 = 6√2. Pull out squared factors in pairs; whatever remains stays under the radical.

What is the difference between a square root and a cube root?

A square root undoes squaring (× itself twice); a cube root undoes cubing (× itself three times). ∛27 = 3 because 3 × 3 × 3 = 27.

Why is √2 called irrational?

Because its decimal never terminates and never repeats — it cannot be written as a fraction of two whole numbers. It is exactly √2, and 1.4142 is only an approximation.

How do I estimate a square root without a calculator?

Bracket it between two perfect squares. √90 sits between √81 = 9 and √100 = 10, so the answer is about 9.5 — and 9.5² = 90.25, confirming it.

Can I take the root of a fraction?

Yes: take the root of the numerator and the root of the denominator separately. √(1/4) = √1 ÷ √4 = 1/2, and √(9/16) = 3/4. The rule works because squaring a fraction squares both parts, so rooting one undoes both. If either part is not a perfect square, the answer stays in radical form, like √(2/9) = √2/3.

Is finding a square root the same as halving a number?

No. Halving divides by 2: half of 16 is 8. Rooting undoes squaring: the square root of 16 is 4, because 4 times 4 is 16. The two operations agree only at one number: the square root of 4 is 2, which is also half of 4, and that coincidence causes endless confusion.

What happens when you square a square root?

You get the original number back. The square of the square root of 7 is 7, the square of the square root of 50 is 50, and the square of 6 times the square root of 2 is 36 times 2, which is 72. That round trip is the definition of a root, and it is the check the calculator performs in its check row.

What is the square root of 10,000?

It is 100, because 100 times 100 is 10,000. The pattern scales cleanly: the square root of 100 is 10, the square root of 1,000,000 is 1,000, and the square root of 1 followed by any even number of zeros is 1 followed by half that many zeros.

Why is the square root of a decimal below 1 larger than the decimal?

Because squaring a number below 1 makes it smaller, so the reverse journey must make it larger. The square root of 0.04 is 0.2, which is larger than 0.04, and the square root of 0.81 is 0.9, which is larger than 0.81, since 0.9 times 0.9 is 0.81. Only numbers above 1 shrink when rooted.

Can a square root be a fraction?

Yes, when the number under the root is a perfect-square fraction. The square root of 1 divided by 4 is 1 divided by 2, and the square root of 25 divided by 36 is 5 divided by 6, because 5 divided by 6, squared, is 25 divided by 36. Irrational roots, such as the square root of 2, can never be written as exact fractions, though any decimal approximation of them is technically a fraction.

Disclaimer: This calculator is for education and everyday maths. Results are shown to 8 decimal places; exact radical form is given for square roots of whole numbers.