Simplifying Radicals Without Losing Your Mind: An Algorithmic Approach

Simplifying radicals is one of those topics where students memorize procedures without understanding why. "Find the largest perfect square factor" is the instruction. But why? And what's the algorithm that works for any radical, not just the ones in the textbook?

What simplifying actually means

sqrt(72) simplifies to 6*sqrt(2). Both expressions equal approximately 8.485. The simplified form is preferred because it separates the rational part (6) from the irrational part (sqrt(2)), making further algebraic manipulation easier.

The principle: sqrt(a*b) = sqrt(a) * sqrt(b). If a is a perfect square, sqrt(a) is an integer, and you've "pulled it out" from under the radical.

For sqrt(72): 72 = 36 * 2. sqrt(36 * 2) = sqrt(36) * sqrt(2) = 6 * sqrt(2).

The algorithm

The textbook approach, "find the largest perfect square factor," works but doesn't scale well. For sqrt(7056), can you identify the largest perfect square factor by inspection? Most people can't.

The algorithmic approach uses prime factorization:

1. Factor the number under the radical into primes: 72 = 2^3 * 3^2
2. For each prime factor, divide the exponent by the radical index (2 for square root, 3 for cube root)
3. Factors with exponent >= index come outside; remainders stay inside

For sqrt(72):

• 72 = 2^3 * 3^2
• For the 2^3: 3 / 2 = 1 remainder 1. One 2 comes outside, one 2 stays inside.
• For the 3^2: 2 / 2 = 1 remainder 0. One 3 comes outside, nothing stays inside.
• Outside: 2 * 3 = 6. Inside: 2.
• Result: 6*sqrt(2).

For cube roots, the same algorithm works with index 3:

• cbrt(108) = cbrt(2^2 * 3^3) = 3 * cbrt(2^2) = 3*cbrt(4)

Rationalizing denominators

A related technique: removing radicals from denominators. 1/sqrt(2) becomes sqrt(2)/2 by multiplying top and bottom by sqrt(2).

For expressions like 1/(sqrt(3) + sqrt(2)), you multiply by the conjugate: (sqrt(3) - sqrt(2))/(sqrt(3) - sqrt(2)). The denominator becomes 3 - 2 = 1, and the result is sqrt(3) - sqrt(2).

This seems like mathematical busywork until you need to add fractions with radical denominators. Common denominators require rationalized forms.

The computational challenge

For large numbers, the prime factorization step is the bottleneck. Factoring a 20-digit number is computationally expensive. For a student tool, trial division up to the square root of the number is sufficient (and fast for numbers under a billion).

function primeFactors(n) {
  const factors = {};
  for (let d = 2; d * d <= n; d++) {
    while (n % d === 0) {
      factors[d] = (factors[d] || 0) + 1;
      n /= d;
    }
  }
  if (n > 1) factors[n] = 1;
  return factors;
}

I built a radical simplifier at zovo.one/free-tools/radical-simplifier that handles square roots, cube roots, and nth roots, with step-by-step prime factorization and simplification. It shows the work so you can follow the algorithm and eventually do it by hand.

I'm Michael Lip. I build free developer tools at zovo.one. 500+ tools, all private, all free.