With 27/64 and index 3 the calculator gives 3/4. With index 2 the same fraction gives √(27/64) ≈ 0.649519052838 (error less than 10⁻¹²), because 27 is not a perfect square. If what you want is to remove the root from the denominator, go to rationalizing denominators.
Worked example: the square root of 8/18
A square tile has an area of 8/18 square foot. How long is its side?
- Simplify the fraction. The greatest common divisor (GCD) of 8 and 18 is 2, so 8/18 = 4/9.
- Take the root of each part. √4 = 2 and √9 = 3.
- Combine the two roots. √(8/18) = √(4/9) = 2/3.
The side is 2/3 ft long.
Check. (2/3)² = 4/9 = 8/18, which is the starting area.
Neither 8 nor 18 is a perfect square, but 8/18 does simplify to one. That is why the calculator simplifies the radicand before looking for the root. If you need to review how to raise fractions to a power, the fraction powers page lets you run the reverse check.
When is the root of a fraction rational?
It depends on the numerator and the denominator separately, once simplified. If both are perfect powers of the index, the root is a fraction. If either one fails, the root is irrational, even though the radicand is rational. Wikipedia states that the square roots of positive integers are irrational unless the integer is a perfect square (accessed October 17, 2026).
| Radicand and index | Result | Why |
|---|---|---|
| 49/64, index 2 | 7/8 | 7² = 49 and 8² = 64 |
| 27/125, index 3 | 3/5 | 3³ = 27 and 5³ = 125 |
| 27/64, index 2 | ≈ 0.649519052838 (error < 10⁻¹²) | 27 is not a perfect square |
| −64/125, index 3 | −4/5 | (−4)³ = −64 and 5³ = 125 |
The radicand 27/64 has a rational cube root and an irrational square root. The index decides, not just the fraction.
Irrational roots and approximate decimals
When the root is not rational, the calculator does not turn it into a fraction. It shows the radical and an approximation marked with ≈. The value is truncated toward zero; with 4 decimals, √(2/3) appears as ≈ 0.8164 (error less than 0.0001), and with 12 decimals as ≈ 0.816496580927 (error less than 10⁻¹²). Rounded to 4 places it would be ≈ 0.8165 (error less than 0.00005), because the next digit is 9. Asking for more decimals shows more digits, but it does not make the root rational.
Many textbooks write the result with no radical in the denominator. For √(2/3), you multiply top and bottom by √3, which gives √(2/3) = √2/√3 = √6/3. It is the same number written differently, because squaring gives (√6/3)² = 6/9, which returns the radicand. The calculator leaves the root as you entered it; the full procedure is in the guide to rationalizing denominators.
If you need a simple fraction near an irrational root, use approximate a decimal as a fraction with a maximum denominator. It will be a nearby fraction, never equal to the root. The article on why the square root of 2 is not a fraction proves the best-known case.
Roots of negative numbers
A negative radicand has a real root only with an odd index, and then it keeps the sign. With an even index, such as √(−1/4), the calculator shows a domain error and does not return complex numbers. The fraction expression calculator has no root operator either, so find the root here and, if it is rational, type the fraction into the expression.
The definition of the principal root appears in Wikipedia’s Square root article, which says that every nonnegative real number has a unique nonnegative square root, called the principal square root (accessed October 17, 2026).
Frequently asked questions
How do I enter a decimal or a mixed number as the radicand?
Type the decimal with a point, or the mixed number with a space. The calculator converts them to a fraction before taking the root: 0.36 is 9/25 and its square root is 3/5; 2 1/4 is 9/4 and its root is 3/2. Repeating decimals, such as 0.(1), are also accepted.
Why is the root of a fraction less than 1 larger than the fraction itself?
Because squaring a number between 0 and 1 makes it smaller. That is why √(1/9) = 1/3, and 1/3 is larger than 1/9. With fractions greater than 1 it goes the other way: √(16/9) = 4/3, which is less than 16/9.