A radical is simplified when no perfect-square factor remains under the root. The job is to find the largest perfect square hiding inside the number and take its square root outside.
Students who know the perfect squares up to 15² = 225 by heart finish these problems in seconds; the rest of the method is a single product rule: √(a·b) = √a · √b.
How to solve it, step by step
Find the largest perfect-square factor
For √72, the perfect squares that divide 72 are 4, 9, and 36. Use 36.
Split the radical
√72 = √(36 · 2) = √36 · √2.
Take the square root of the perfect square
√36 = 6, so √72 = 6√2.
Handle variables by pairs
√(x⁵) = √(x⁴ · x) = x²√x. Every pair of the variable comes out as one factor.
Worked example
Example 1
Simplify √200.
- Largest perfect square factor of 200: 100
- √200 = √(100 · 2) = 10√2
Answer 10√2
Example 2
Simplify 3√(48x³y⁶).
- 48 = 16 · 3; x³ = x² · x; y⁶ = (y³)²
- √(48x³y⁶) = 4 · x · y³ · √(3x)
- Multiply by the 3 in front: 12xy³√(3x)
Answer 12xy³√(3x)
Mistakes DeltaMath marks wrong
- Stopping at a smaller perfect square: √72 = 2√18 is not simplified because 18 still contains 9.
- Adding instead of multiplying: √(36 + 36) is not 6 + 6.
- Pulling a variable out without halving its exponent.
- Typing 6sqrt2 without parentheses so the answer box reads it as 6·sqrt(2) only by luck; use sqrt(2).
How to type the answer
- Type sqrt() around the radicand: 6sqrt(2), 12xy^3sqrt(3x).
- The math keyboard's √ button inserts the same structure.
- Do not convert to decimals unless the problem asks you to round.