Square Root Calculator
The square root calculator gives the simplest radical form as well as the decimal: the square root of 72 is 6 times the root of 2, or 8.485281374.
Root
- Simplest radical form 6 squared times 2 is 72.
- 6sqrt(2)
- How that comes out 2^3 gives 2^1 outside; 3^2 gives 3^1 outside.
- 2^3 x 3^2
- Both roots
- +8.485281374 and -8.485281374
- Between Because 8^2 = 64 and 9^2 = 81, and 72 sits between them.
- 8 and 9
- Check
- 8.485281374^2 = 72
- As a power
- 72^(1/2)
- Reciprocal 1 / sqrt(72), sometimes written sqrt(72)/72 to rationalize the denominator.
- 0.1178511302
| n | sqrt(n) | Radical form |
|---|---|---|
| 69 | 8.3066239 | sqrt(69) |
| 70 | 8.3666003 | sqrt(70) |
| 71 | 8.4261498 | sqrt(71) |
| 72 | 8.4852814 | 6sqrt(2) |
| 73 | 8.5440037 | sqrt(73) |
| 74 | 8.6023253 | sqrt(74) |
| 75 | 8.660254 | 5sqrt(3) |
How the Square Root Calculator works
A square root of a number is a value that, multiplied by itself, gives that number. The square root of 9 is 3, because 3 x 3 = 9.
Most numbers are not perfect squares, and their square roots are irrational: the decimal never terminates and never repeats, so any decimal is a rounding. The exact answer comes from prime factorization. Every pair of identical factors inside a square root comes out as one copy of that factor, and the radicand that remains is square-free by construction, which is what "simplest radical form" means: there is nothing left to pull out.
The same procedure explains which roots are whole numbers. If every prime appears an even number of times, everything comes out and the radicand is 1: 144 = 2^4 x 3^2, so sqrt(144) = 12 exactly. Any odd exponent leaves something inside and the root is irrational. There is no middle case: the square root of an integer is either an integer or irrational, never a fraction. The result goes back to the Greeks and was reportedly unwelcome at the time.
For mental estimates, bracketing is the fastest route. sqrt(72) sits between 8 and 9 because 8^2 = 64 and 9^2 = 81. 72 is closer to 64, so the answer is a bit under the midpoint, around 8.5. One step of Newton's method sharpens it: average your guess with the number divided by the guess. Starting at 8.5, (8.5 + 72/8.5) / 2 = 8.4853, which is right to four decimal places from one iteration.
Higher roots follow the same logic with different thresholds. A cube root pulls out groups of three, so the cube root of 54 = 2 x 3^3 is 3 times the cube root of 2. The general definition is that the nth root of x is x raised to the power 1/n, which is what lets non-integer roots exist at all.
Square Root Calculator FAQ
What is the square root of 72 in simplest radical form?
6sqrt(2). Factorize 72 as 2^3 x 3^2, then pull out every pair: the 3^2 gives a 3 and two of the 2s give a 2, so 6 comes out and a single 2 stays in. As a decimal it is 8.485281..., which is irrational and so never exact.
How do I simplify a square root?
Find the prime factorization and take out one copy of every pair. sqrt(180): 180 = 2^2 x 3^2 x 5, so the 2^2 gives a 2 and the 3^2 gives a 3, leaving 6sqrt(5). Equivalently, find the largest perfect square that divides the number (36 here) and take its root outside.
Does a positive number have two square roots?
Yes. 9 has roots 3 and -3, since both square to 9. The radical symbol denotes the positive one by convention, which is why solving x squared equals 9 needs an explicit plus-or-minus while sqrt(9) alone means 3.
Can you take the square root of a negative number?
Not within the real numbers, because any real number squared is non-negative. In the complex numbers sqrt(-9) is 3i, where i is defined as sqrt(-1). Odd roots are different: the cube root of -8 is -2, which is a perfectly ordinary real number.