Math
Radical Simplifier
Simplify any nth root of a positive integer. Pulls every perfect nth-power factor out of the radical and shows the prime factorization, coefficient, and decimal value.
Simplify a radical
Integer from 1 to 1,000,000.
Pick the root degree (2 to 10).
Simplified form
√75 = 5√3
Pulled 5 outside the radical; 3 stays inside.
Coefficient
5
Remaining radicand
3
Decimal value
8.660254
Prime factorization of 75
3×52
Frequently Asked Questions about the Radical Simplifier
What does it mean to simplify a radical?
Simplifying a radical means rewriting an nth root of an integer in the form a * nth-root(b), where b has no perfect-nth-power factors left inside. The numeric value does not change; the expression just becomes shorter and easier to work with. Square root of 75 simplifies to 5 * square root of 3 because 75 = 25 * 3 and 25 is a perfect square.
How does prime factorization help simplify radicals?
Once you write the radicand as a product of prime powers, the rule is mechanical: for each prime p with exponent e, pull floor(e / index) copies of p out of the radical and leave e mod index copies inside. For the cube root of 54, the factorization is 2 * 3^3, so the three 3s pull out as a single 3 and the 2 stays inside, giving 3 * cube root of 2.
Why does square root of 75 equal 5 times square root of 3, not 5 times 3 under the radical?
The two are different expressions. 5 * square root of 3 means 5 multiplied by the irrational number sqrt(3), about 8.660. 5 * sqrt(3) under a single radical would mean sqrt(5 * 3) = sqrt(15), about 3.873. The coefficient that gets pulled out sits outside the radical sign as a regular multiplier; only the leftover factor stays inside.
Does this work for cube roots and higher?
Yes. The same rule applies for any index from 2 to 10. For a cube root you pull out copies of a prime in groups of three; for a fourth root, groups of four; for a fifth root, groups of five, and so on. The cube root of 8 is 2 because 8 = 2^3 (the three 2s pull out as a single 2 and nothing stays inside). The fifth root of 32 is 2 for the same reason: 32 = 2^5.
When is a radical already in simplest form?
A radical is fully simplified when its radicand has no perfect nth-power factor other than 1. That happens whenever the radicand is prime (sqrt(2), sqrt(7), cube root of 5), or when every prime in its factorization has an exponent smaller than the index. Square root of 30 is already simplest because 30 = 2 * 3 * 5 and no prime appears twice. The calculator returns coefficient 1 in those cases.