Simplify Radical Calculator

Simplifying a radical means rewriting a square root in its simplest form by extracting any perfect square factors.

The Process:

1. Find the prime factorization of the number under the radical.
2. Group the prime factors into pairs.
3. For each pair, pull one copy outside the radical.
4. Whatever remains inside stays under the radical sign.

Worked Example: sqrt(72)

1. Prime factorization: 72 = 2³ x 3² = 2 x 2 x 2 x 3 x 3
2. Pairs: one pair of 2s, one pair of 3s, one leftover 2
3. Pull out: 2 x 3 = 6
4. Remaining inside: 2
5. Result: sqrt(72) = 6sqrt(2)

Check: 6² x 2 = 36 x 2 = 72.

Another Example: sqrt(200)

200 = 2³ x 5² = (2²)(5²)(2). Pull out 2 x 5 = 10, leave 2 inside. sqrt(200) = 10sqrt(2).

Perfect Squares:

If all prime factors come in pairs, the number is a perfect square and the radical simplifies to a whole number. sqrt(144) = 12 because 144 = 2⁴ x 3² = (2² x 3)² = 12².

Why simplify radicals? Simplified form makes it easier to compare, add, and multiply radical expressions. In algebra, teachers expect answers in simplified radical form rather than as decimals.