Calculate square roots, cube roots, and nth roots instantly. Find any root of any number with step-by-step solutions. Free online root calculator with radical form, decimal approximation, and formula explanations.
In mathematics, the general root, or the nth root of a number a is another number b that when multiplied by itself n times, equals a:
Think of it like this. You know how multiplication is just repeated addition? Like 3 x 4 is just 3 + 3 + 3 + 3?
Well, roots are the opposite of exponents. An exponent says "multiply this number by itself this many times." So 3² means 3 x 3 = 9.
A root asks the reverse question. "What number, multiplied by itself a certain number of times, gives me this?"
So the square root of 9 is 3. Because 3 x 3 = 9.
The cube root of 27 is 3. Because 3 x 3 x 3 = 27.
See the pattern?
Most people know square roots. That's the one with the little checkmark symbol: √.
A square root asks: "What number times itself equals this?"
A cube root asks: "What number times itself three times equals this?" It looks like this: ∛.
An nth root is just the general version. You can have a 4th root, a 5th root, a 10th root. Any number works.
Our calculator handles all of them. Just type in the "n" for the root you want.
It's super simple. Here's the step-by-step:
For 64 with a square root? You get 8. Because 8 x 8 = 64.
For 64 with a cube root? You get 4. Because 4 x 4 x 4 = 64.
For 64 with a 6th root? You get 2. Because 2 x 2 x 2 x 2 x 2 x 2 = 64.
Try it yourself. It's that easy.
Some roots show up all the time. Memorizing these will save you a ton of time.
Notice a pattern? The cube root of 8 is 2. The cube root of 27 is 3. The numbers are getting bigger, but the roots are getting smaller. That's because cubes grow fast.
This is where it gets interesting. Can you find the square root of -9?
Think about it. What number times itself equals -9?
3 x 3 = 9. Not -9.
-3 x -3 = 9. Also not -9.
There's no real number that works. That's why your calculator says "Error" or "Undefined" when you try.
But cube roots are different. The cube root of -8 is -2. Because -2 x -2 x -2 = -8.
So here's the rule: Even roots (square, 4th, 6th) of negative numbers don't exist in the real world. Odd roots (cube, 5th, 7th) do.
Our calculator handles this correctly. Try it.
The square root of 0 is 0. The cube root of 0 is 0. Any root of 0 is 0.
Simple enough.
To calculate √a, use the following iterative method:
Example: Find √27 to 3 decimal places
Guess: 5.125 → 27÷5.125=5.268 → avg=5.197
27÷5.197=5.195 → avg=5.196 → 27÷5.196=5.196 ✓
√27 ≈ 5.196
For higher roots, the method is similar but accounts for the degree n:
Example: Find ⁸√15 to 3 decimal places
Guess: 1.432 → 15÷1.432⁷=1.405 → avg=(1.432×7+1.405)/8=1.388
15÷1.388⁷=1.403 → avg=(1.403×7+1.388)/8=1.402
⁸√15 ≈ 1.402
Sometimes you don't have a calculator handy. Or you want to understand the math better. Here's how to do it by hand.
Let's find √20. You know √16 = 4 and √25 = 5. So √20 is between 4 and 5.
Try 4.5. 4.5 x 4.5 = 20.25. Too high.
Try 4.4. 4.4 x 4.4 = 19.36. Too low.
Try 4.47. 4.47 x 4.47 = 19.98. Close enough.
So √20 ≈ 4.47.
This works great for perfect squares. Let's find √144.
First, break 144 into prime factors: 144 = 2 x 2 x 2 x 2 x 3 x 3.
Now group them in pairs: (2 x 2) x (2 x 2) x (3 x 3).
Take one number from each pair: 2 x 2 x 3 = 12.
So √144 = 12.
This is an ancient trick that still works. Let's find √10.
Step 1: Make a guess. Let's say 3.
Step 2: Divide 10 by your guess. 10 ÷ 3 = 3.33.
Step 3: Average your guess and the result. (3 + 3.33) ÷ 2 = 3.165.
Step 4: That's your new guess. Repeat.
10 ÷ 3.165 = 3.159. Average: (3.165 + 3.159) ÷ 2 = 3.162.
One more time: 10 ÷ 3.162 = 3.162. They match. So √10 ≈ 3.162.
Pretty cool, right?
You might be thinking, "When will I ever use this?" More often than you'd think.
Need to build a square deck? The diagonal length uses square roots. If your deck is 10 feet by 10 feet, the diagonal is √(10² + 10²) = √200 ≈ 14.14 feet.
Compound interest calculations use roots. If you want to double your money in 10 years, you need an annual return of about 7.2%. That comes from a root calculation.
Speed, acceleration, and force all use roots. The time it takes for an object to fall from a height uses the square root of the height.
Ever played a video game? The distance between two points on the screen uses square roots. It's called the distance formula.
This is a big one for students. Let's say you need to simplify √12.
Step 1: Find the largest perfect square that divides 12. That's 4.
Step 2: Write 12 as 4 x 3.
Step 3: √12 = √(4 x 3) = √4 x √3 = 2√3.
That's it. √12 simplified is 2√3.
Try another: √50. The largest perfect square is 25. So √50 = √(25 x 2) = 5√2.
One more: √72. Largest perfect square is 36. So √72 = √(36 x 2) = 6√2.
See the pattern? Find the biggest perfect square factor, pull it out, and leave the rest inside.
This blew people's minds in ancient Greece. √2 is about 1.414, but the decimal never ends and never repeats. It's irrational.
Here's the simple proof. If √2 were rational, you could write it as a fraction a/b where a and b are whole numbers with no common factors.
Then (a/b)² = 2, so a² = 2b². This means a² is even, so a is even. Write a = 2k.
Then (2k)² = 2b², so 4k² = 2b², so 2k² = b². This means b² is even, so b is even.
But if a and b are both even, they have a common factor of 2. That contradicts our assumption. So √2 can't be rational.
Mind-blowing, right?
When you solve x² = 9, x can be 3 or -3. But √9 is only 3. The square root symbol means the principal (positive) root.
√4 + √9 is not √13. It's 2 + 3 = 5. You can't add roots directly unless they're the same.
√2 x √3 = √6. That's correct. But √2 x √2 = 2, not √4. Because √2 x √2 = (√2)² = 2.
The square root symbol (√) comes from the Latin word "radix," meaning root. It was first used in the 16th century by a German mathematician named Christoph Rudolff.
Before that, people wrote "R" for root. The symbol we use today is a stylized version of that letter.
Ancient Babylonians had a method for finding square roots that's almost identical to the Babylonian method we showed earlier. They were doing this 4,000 years ago.
Our root calculator works on any device. Phone, tablet, laptop. It's all good.
You can enter decimals too. Need the square root of 2.5? Just type it in.
The calculator handles huge numbers. Try the square root of 1,000,000. It's 1,000.
And it's completely free. No ads, no sign-ups, no nonsense.
A square root asks what number times itself equals the original number. A cube root asks what number times itself three times equals the original number. So the square root of 8 is about 2.83, but the cube root of 8 is exactly 2.
Because there's no real number that, when multiplied by itself, gives a negative result. A positive times a positive is positive. A negative times a negative is also positive. So square roots of negative numbers don't exist in the real number system.
You can use the guess and check method, prime factorization, or the Babylonian method. The Babylonian method is the most accurate. Make a guess, divide the number by your guess, average the two, and repeat until the answer stabilizes.
The square root of 0 is 0. Any root of 0 is 0. Because 0 times itself any number of times is still 0.
Yes. Just enter the number and set the root to 3. The calculator will give you the cube root instantly. It works for any root, not just square roots.
Simplifying a square root means finding the largest perfect square factor and pulling it out. For example, √12 simplifies to 2√3 because 12 = 4 x 3 and √4 = 2. The simplified form is easier to work with in equations.