What is a "simplified radical"?
A simplified radical means the number under the square root (the radicand) has no perfect-square factors other than 1. In other words, you have taken out any whole-number square roots you can. Often it also means we avoid having a square root in the denominator, but the main idea is: no perfect squares left inside the root.
Rules (short)
- If a number inside the root can be written as a perfect square times another number, take the perfect square out: sqrt(a * b) = sqrt(a) * sqrt(b).
- Use prime-factor pairs: each pair of the same prime comes out of the root as one of that prime.
- Stop when the number left inside the root has no perfect square factors (other than 1).
Step-by-step: simplify sqrt(24)
- Find a perfect square factor of 24. One easy one is 4 because 24 = 4 * 6.
- Use sqrt(ab) = sqrt(a) * sqrt(b): sqrt(24) = sqrt(4 * 6) = sqrt(4) * sqrt(6).
- Sqrt(4) is 2, so sqrt(24) = 2 * sqrt(6) = 2sqrt6.
- Check sqrt(6): 6 = 2 * 3 and has no perfect-square factors, so it cannot be simplified further. Therefore 2sqrt6 is fully simplified.
Prime-factor method (another way)
Factor 24 into primes: 24 = 2 * 2 * 2 * 3 = 2^3 * 3. Pair up matching primes: one pair of 2s (2^2) comes out as a single 2, leaving one 2 and the 3 inside: sqrt(24) = 2 * sqrt(2 * 3) = 2sqrt6. Same result.
More examples
- sqrt(75) = sqrt(25 * 3) = 5sqrt3
- sqrt(18) = sqrt(9 * 2) = 3sqrt2
- sqrt(48) = sqrt(16 * 3) = 4sqrt3 (or prime method: 48 = 2^4 * 3 -> two pairs of 2 give 4)
Practice (try them, then check answers)
- sqrt(32) = ? Answer: 4sqrt2
- sqrt(40) = ? Answer: 2sqrt10
- sqrt(45) = ? Answer: 3sqrt5
If you want, I can give more practice problems or explain rationalizing the denominator (how to remove radicals from bottoms of fractions).