Square Root Calculator
Find square roots, cube roots and nth roots up to n = 20, with the simplified radical form for whole numbers and a check that raises the root back to the power n.
What is a square root calculator?
The square root of a number x is the number that, multiplied by itself, gives x: √49 = 7 because 7 × 7 = 49. More generally, the nth root of x is the number r with rⁿ = x. The cube root (n = 3) of 125 is 5 because 5 × 5 × 5 = 125.
This square root calculator finds the root of any number for a root index from 2 to 20. Enter the Number and the Root index (n): 2 for a square root, 3 for a cube root, 4 for a fourth root, and so on. It returns:
- nth root: the value to 8 decimal places.
- Simplified radical: for whole numbers, the exact form with every perfect nth-power factor taken out, such as √72 = 6√2 or ∜1250 = 5∜2.
- Perfect root?: Yes when the root is a whole number, as in ∛(-27) = -3.
- Check: the root raised back to the power n, so you can see it returns your number.
Odd roots of negative numbers are real and negative, so the calculator gives ∛(−27) = −3. Even roots of negative numbers, such as √(−16), are not real numbers, and the calculator says so instead of showing an error.
The Common Core grade 8 standard 8.EE.A.2 asks students to use square root and cube root symbols, evaluate roots of small perfect squares and cubes, and know that √2 is irrational. In high school, roots are rewritten as rational exponents (N-RN.A.1 and N-RN.A.2), so ⁿ√x = x1/n.
Formula
The nth root is the inverse of raising to the power n:
ⁿ√x = r means rⁿ = x, and in exponent form ⁿ√x = x1/n.
- Positive x: the calculator gives the positive root, called the principal root. Both 4 and −4 square to 16, but √16 means 4.
- Negative x, odd n: the root is negative, because an odd number of negative factors is negative: (−3)³ = −27, so ∛(−27) = −3.
- Negative x, even n: no real number works, because any real number raised to an even power is 0 or positive.
Simplifying a radical
To simplify ⁿ√x, write x as kⁿ × m, where kⁿ is the largest perfect nth power that divides x. Then
ⁿ√(kⁿ × m) = k × ⁿ√m
For example, 72 = 36 × 2 = 6² × 2, so √72 = 6√2. The calculator finds k by testing 2, 3, 4 and so on, dividing out each perfect nth power it finds. If nothing is left under the radical (m = 1), x is a perfect nth power and the simplified form is just a whole number.
Perfect powers to recognize
| k | k² (perfect square) | k³ (perfect cube) | k⁴ (perfect fourth power) |
|---|---|---|---|
| 2 | 4 | 8 | 16 |
| 3 | 9 | 27 | 81 |
| 4 | 16 | 64 | 256 |
| 5 | 25 | 125 | 625 |
| 6 | 36 | 216 | 1,296 |
| 7 | 49 | 343 | 2,401 |
| 8 | 64 | 512 | 4,096 |
| 9 | 81 | 729 | 6,561 |
| 10 | 100 | 1,000 | 10,000 |
Any whole number that is not a perfect nth power has an irrational nth root: its decimal never ends or repeats, so the 8-decimal value is a rounded approximation and the simplified radical is the exact answer.
Worked example
Example 1: simplify √72
- Find the largest perfect square that divides 72. Try 36: 72 ÷ 36 = 2, so 72 = 36 × 2.
- √72 = √36 × √2 = 6√2, which is what Simplified radical shows.
- nth root gives 8.48528137, and Check reads
8.48528137² = 72. Perfect root? is No.
A common mistake is to stop at a smaller square: 72 = 4 × 18 gives 2√18, which is not fully simplified because 18 still contains the square 9.
Example 2: cube roots of negative numbers
Enter −27 with n = 3. The nth root is −3, Simplified radical shows −3, Perfect root? reads Yes (∛(−27) = −3), and Check reads (−3)³ = −27.
Now try −16 with n = 3. Since −16 = −(8 × 2) and 8 = 2³, the simplified radical is −2∛2, about −2.5198421. Change n to 2 and the result changes completely: Simplified radical reads "Not a real number (even root of a negative number)".
Example 3: a fourth root
Enter 1250 with n = 4.
- Look for a perfect fourth power that divides 1250. 5⁴ = 625, and 1250 ÷ 625 = 2.
- ∜1250 = ∜625 × ∜2 = 5∜2.
- The decimal value is 5.94603558, and Check reads
5.94603558^4 = 1250.
Example 4: a decimal input
Enter 2.25 with n = 2. The nth root is 1.5 and Check reads 1.5² = 2.25. Simplified radical shows a dash, because radicals are simplified only for whole numbers, and Perfect root? says No, because it only answers Yes when the root is a whole number. 2.25 is still the square of a fraction: 1.5 = 3/2, and (3/2)² = 9/4 = 2.25.
How to use this calculator
- Enter the Number you want the root of. Negative numbers and decimals are allowed.
- Enter the Root index (n): 2 for a square root, 3 for a cube root, or any whole number up to 20.
- Read the decimal value in nth root.
- For an exact answer, use Simplified radical; this is usually the form a teacher wants when the root is irrational.
- Use Perfect root? and Check to confirm the answer.
Common mistakes
- Splitting a sum. √(9 + 16) = √25 = 5, not √9 + √16 = 7. Roots split over multiplication, not addition.
- Not simplifying fully, as in 2√18 instead of 6√2.
- Forgetting the negative solution. √16 is 4, but the equation x² = 16 has two solutions, 4 and −4.
- Taking an even root of a negative number. √(−16) is not a real number; ∛(−16) is.
Limits of the calculator
- The root index must be a whole number from 2 to 20. A blank index counts as 2, a decimal index is rounded down (2.7 becomes 2), and values outside the range are moved to 2 or 20.
- Simplified radical works for whole numbers up to one trillion; decimals and larger numbers show a dash.
- For index 5 and above, the index is written as a small raised number in front of √, so ⁵√ means "fifth root". When there is also a whole-number factor, a dot separates the two: 96 with n = 5 shows 2·⁵√3, which means 2 × ⁵√3.
- Simplified radical covers whole numbers up to one trillion (10¹²) and shows a dash above that. Perfect root? uses exact whole-number arithmetic up to about 9 × 10¹⁵, so it does not mistake a number 1 away from a perfect power for a perfect power.
For the methods behind the answers, read how to reduce square roots to lowest form and how to find square roots by hand. Square roots also appear in the Pythagorean theorem calculator and the quadratic equation solver.
Sources
Frequently asked questions
Why is the square root of a negative number "not a real number"?
Any real number times itself is 0 or positive: 4 × 4 = 16 and (−4) × (−4) = 16. So no real number squares to −16. In the complex numbers it does have square roots, √(−16) = 4i, where i² = −1, but this calculator works with real numbers only. The same applies to every even root (4th, 6th and so on) of a negative number.
Why does √16 give only 4 and not −4?
The √ symbol means the principal (non-negative) square root, so √16 = 4. The equation x² = 16 is a different question and has two answers, x = 4 and x = −4, often written x = ±4. When you solve a quadratic, you add the ± yourself; the quadratic equation solver does this for you.
What counts as a perfect root?
A number is a perfect nth power when its nth root is a whole number: 144 is a perfect square (√144 = 12), 343 is a perfect cube (∛343 = 7), and 1024 is a perfect fifth power (the fifth root is 4). The Perfect root? line follows this definition, so it says No for 2.25 even though √2.25 = 1.5 exactly. A root that is not a whole number may be a terminating decimal, like 1.5, or irrational, like √2.
How does the calculator simplify a radical?
For a whole number up to one trillion, it tests k = 2, 3, 4 and so on, and each time kⁿ divides the number, it moves a factor of k outside the radical. Whatever is left stays under the radical. For √75 it finds 25 = 5², so √75 = 5√3; for ∛40 it finds 8 = 2³, so ∛40 = 2∛5; and for ∜162 it finds 81 = 3⁴, so ∜162 = 3∜2.
How accurate is the decimal answer?
The nth root is shown to 8 decimal places, which is more than school and most practical work needs. It comes from ordinary computer floating-point arithmetic, so the last digit can be off by a tiny amount. For a perfect power, the Check line uses the exact whole-number root instead, so the cube root of 1,000,000,000 checks as 1000³ = 1000000000. For other numbers it raises the rounded root to the power, as in 8.48528137² = 72. Use Simplified radical when you need an exact answer.
Is a square root the same as raising to the power ½?
Yes. For non-negative x, √x = x1/2, and in general ⁿ√x = x1/n. The Common Core standard N-RN.A.1 explains why: 51/3 is defined as the cube root of 5 so that (51/3)³ = 5 follows the usual exponent rules. The exponent calculator gives 720.5 the same value as √72 here, 8.48528137. For a negative base, use this calculator with an odd index instead, because the exponent calculator does not accept a negative base with a non-whole exponent.
Can I find the 25th root or a root with a decimal index?
No. The Root index (n) is limited to whole numbers from 2 to 20. A decimal index is rounded down and an index above 20 is treated as 20; the Check line shows the index actually used, such as 1.2384162^20 = 72 when 25 is entered for 72. For other roots, use the exponent calculator with the matching power: the 25th root of 72 is 720.04 = 1.18656983.