Square Root Calculator
Enter a number and a root degree — 2 for a square root — and get the root, whether the number is a perfect square, and the radical in its simplest form, so 72 comes back as 6√2.
Odd roots of negative numbers are real numbers: the cube root of -8 is -2. Even roots of negative numbers are not.
Formula and calculation
Worked example
The square root of 72 is 8.4853, and as a radical it simplifies to 6√2, because 72 = 36 × 2 and 36 is 6². Squaring the answer gives 72, the check that nothing slipped, and the perfect-square test reports No — the nearest squares are 8² = 64 and 9² = 81, so 72 is not a perfect square.
How a radical gets simplified
Splitting a whole number into a square factor and the rest is the whole trick: 8 = 4 × 2 gives 2√2, 50 = 25 × 2 gives 5√2, and 72 = 36 × 2 gives 6√2. The decimal value is unchanged — 6√2 is 8.485281… — only the notation gets tidier. Factors up to 10,000 are searched, which covers every square factor of numbers up to nine digits in almost every case that matters in school work. Beyond that bound a larger factor could exist and the form shown may not be fully reduced. Related: an exponent of 1/2 is the same operation, and the exponent calculator handles it from the power side.
Roots, powers and logarithms
Roots, powers and logarithms answer three versions of one question. A power is told the base and the exponent and asks for the result: 10³ = 1,000. A root is told the result and the power and asks for the base: the cube root of 1,000 is 10. A logarithm is told the base and the result and asks which exponent did it, so log10(1000) = 3. That is why a root can be written as a fractional exponent and why logs undo powers — three views of the same relationship.
What the results mean
- Result
- The n-th root of the number — the value that, multiplied by itself n times, gives the number back. The square root of 144 is 12, and the cube root of −8 is −2.
- Result squared
- The result raised back to the power n: a precision check. For a square root it is the result squared and it should match the number you entered.
- Perfect square
- Whether the number is a perfect square (or cube, or n-th power) — a whole number produced by raising a whole number to that power. 144 = 12² is one; 72 is not, and sits between 8² and 9².
- Simplified form
- The radical in its simplest form: the square root of 72 is 6√2 because 72 = 36 × 2 and the square root of 36 is 6. Factors are searched up to 10,000 (k^n ≤ 10,000), so a very large number may still have a bigger factor to pull out.
- Calculation
- The arithmetic in words, including the factorization behind the simplified form: the square root of 72 is 6√2 ≈ 8.485281 because 72 = 36 × 2.
Frequently asked questions
How do I simplify the square root of 72?
Find the largest square number that divides it: 72 = 36 × 2, and 36 is 6², so the square root of 72 is 6√2, which is 8.485281…. This calculator pulls out the largest square factor up to 10,000 for any whole number you enter.
What is the square root of a negative number?
There is no real answer, because no real number squares to a negative. The square root of −4 is 2i in the complex numbers. Odd roots are different: the cube root of −8 is −2, since (−2)³ = −8, so set the root degree to 3 and enter −8.
How do I work out a cube root or a fifth root?
Change the root degree: 2 is a square root, 3 a cube root, 4 a fourth root, up to 1000. The cube root of 27 is 3 and the fifth root of 32 is 2. The number 2.5 with a degree of 2 is fine too, though the simplified radical form only applies to whole numbers.
Is the square root of 2 a rational number?
No. Its decimal 1.41421356… never ends and never repeats, so it cannot be written as a fraction of whole numbers — that is what irrational means. Numbers with whole-number roots, such as 144, 169 and 2,500, are called perfect squares instead.
How accurate is the root this calculator shows?
The value is held at full double precision, about 15 significant digits, and only the display is rounded. Squaring the result is the honest check: root 2 shown as 1.414214 comes back as 2.0000000000000004 when multiplied by itself, which is as close as a computer gets.