HesapHanem

Radical Expression Simplifier

Enter the index and radicand to simplify the radical.

Result

6√5
6
5

How it is calculated

Enter the root index n (2 for a square root, 3 for a cube root) and the whole number N under the root. The tool factors N into primes. Each prime with exponent e contributes p^⌊e ÷ n⌋ to the coefficient outside the root and p^(e mod n) to the radicand left inside, so ⁿ√N = coefficient × ⁿ√radicand. The index must be at least 2 and N cannot be negative. Values that are not whole are rounded to the nearest integer (50.7 becomes 51). N = 0 and N = 1 return themselves. Square roots are written with √, cube roots with ∛, and higher roots as coefficient·(radicand)^(1/n). If nothing can be extracted, the coefficient is 1. If the radicand ends as 1, the root is a whole number. Very large N may lose precision, since numbers are held as floating point.

Worked example

Simplify the square root of 180, so n = 2. Step 1: factor, 180 = 2² × 3² × 5. Step 2: each pair of equal primes comes out of the root. The 2² gives a 2, the 3² gives a 3, and the 5 appears once, so it stays inside. Step 3: the coefficient is 2 × 3 = 6 and the radicand is 5. The tool shows 6√5. As a check, 6² × 5 = 36 × 5 = 180.

Try this example →

Common mistakes

  • Typing the radicand in the index field and the index in the radicand field. For the cube root of 54, n is 3 and N is 54; swapped, the tool is asked for the 54th root of 3 and returns (3)^(1/54).
  • Expecting a decimal. The result is an exact form such as 6√5, not 13.416; if you need a decimal, work it out from the simplified form separately.
  • Entering a decimal or a fraction under the root. N is rounded to a whole number, so 50.7 becomes 51 and 1.5 becomes 2 without any warning.
  • Entering a negative number. A negative radicand returns no result; this tool does not handle imaginary numbers.
  • Expecting a prime or an already simplified number to change. √97 and √30 come back as they are, because no square factor can be taken out.

Sources

About this calculator

Simplifying an nth root means factoring the radicand (the number under the root) to pull out any perfect nth powers, leaving the smallest possible number inside the radical. For a square root, this means finding the largest perfect-square factor: √50 = √(25 × 2) = √25 × √2 = 5√2, since 25 is a perfect square but 2 has no square factors left to extract. The same logic extends to cube roots and higher: ∛54 = ∛(27 × 2) = 3∛2, because 27 = 3³ is a perfect cube.

Simplified radical form is the standard way to express irrational answers exactly (rather than as a rounded decimal), which matters for exact arithmetic, further algebraic manipulation, and matching answer keys. Algebra and geometry students, standardized-test takers, and math teachers use this tool to check simplification steps on problems involving the Pythagorean theorem, quadratic formula solutions, or any expression that naturally produces an irrational root.

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Comments (1)

  • Editorial team

    Simplifying is most useful before you add radicals. Run each term through the tool first: the square roots of 12, 27 and 75 come out as 2√3, 3√3 and 5√3, so their sum is 10√3, which you would never spot from the raw numbers. Only terms with the same index and the same radicand can be combined. Which kind of problem sends you here most often, sums of radicals or a quadratic formula answer?

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