What a "simplified radical" means
- The expression under the root (the radicand) has no perfect-square factor greater than 1 (for square roots). For nth roots, it has no perfect n-th power factor greater than 1.
- There is no radical in the denominator of a fraction (you usually "rationalize" the denominator).
- The number outside the radical (the coefficient) is an integer and you've moved out every factor that can be taken out.
Why √24 is not simplified
24 contains a perfect square factor: 4 (because 24 = 4 × 6). Since √(a·b) = √a · √b, you can take √4 out of the radical:
√24 = √(4·6) = √4 · √6 = 2√6.
So √24 is not simplified, and its simplified form is 2√6.
Step-by-step method to simplify a square root
- Factor the radicand into primes (or into known square factors).
- Group prime factors into pairs (for square roots). Each pair can come out of the square root as a single factor.
- Multiply the numbers that came out (that is your coefficient). Multiply the leftover unpaired primes inside the radical to get the new radicand.
Example using prime factors: √24
- Prime factorization: 24 = 2³ · 3.
- For square roots, take groups of two: from 2³ there is one pair of 2 (2²) and one leftover 2. So a factor 2 moves outside.
- Leftover inside: 2 · 3 = 6. So √24 = 2√6.
More examples
- √18 = √(9·2) = 3√2
- √72 = √(36·2) = 6√2
- ∛54 (cube root) = ∛(27·2) = 3∛2 — for nth roots, group powers in n's.
Rationalizing a denominator (common extra step)
If you have a fraction like 3/√2, multiply numerator and denominator by √2 to remove the radical from the denominator:
3/√2 · (√2/√2) = 3√2 / 2.
For denominators with two terms (like a + √b), multiply by the conjugate (a − √b).
Quick checklist to know a radical is simplified
- The radicand has no perfect square factor > 1 (for square roots).
- No radicals in the denominator.
- All possible integer factors have been taken out of the root.
Practice (try these, then check answers)
- Simplify √50 (answer: 5√2)
- Simplify √200 (answer: 10√2)
- Rationalize: 1/√3 (answer: √3/3)
If you want, I can give more practice problems, show how to handle higher roots in detail, or help with rationalizing denominators that are binomials.