How do you simplify square roots?
Split the number into a perfect square times something else, then take the square root of the perfect square. The product rule √(ab) = √a × √b is what makes this allowed.
Here is √72 worked out step by step. The perfect squares to look for are 4, 9, 16, 25, 36, 49, 64, 81 and 100.
If the radical already has a number in front, multiply it by whatever comes out. For 3√12, write √12 = √4 × √3 = 2√3, so 3√12 = 3 × 2√3 = 6√3.
How do you simplify radicals using prime factors?
Write the number under the root as a product of primes and circle every pair of matching primes. Each pair comes out of a square root as a single number, and anything without a partner stays inside.
For √72, the primes are 2 × 2 × 2 × 3 × 3. There is one pair of 2s and one pair of 3s, with a single 2 left over. So √72 = 2 × 3 × √2 = 6√2, the same answer as before.
Bigger numbers show why this helps. For √200, the primes are 2 × 2 × 2 × 5 × 5. One pair of 2s and one pair of 5s come out, leaving a single 2, so √200 = 2 × 5 × √2 = 10√2.
This is the method to use when you can't spot the biggest perfect square right away. It also explains how to simplify radicals by hand for any size number, since every whole number has a prime factorization.
How do you simplify a cube root?
For a cube root, look for perfect cube factors (8, 27, 64, 125) or groups of three matching primes. Each group of three comes out as one number.
For ∛54, notice that 54 = 27 × 2. Since ∛27 = 3, you get ∛54 = 3∛2. In prime form, 54 = 2 × 3 × 3 × 3, so the three 3s come out as a single 3 and the 2 stays inside. Another example: ∛16 = ∛(8 × 2) = 2∛2.
| Input | Factor used | Simplified |
|---|---|---|
| sqrt(72) | 36 × 2 | 6√2 |
| 3sqrt(12) | 4 × 3 | 6√3 |
| sqrt(45) | 9 × 5 | 3√5 |
| cbrt(54) | 27 × 2 | 3∛2 |
| cbrt(16) | 8 × 2 | 2∛2 |
How do you rationalize a denominator?
Multiply the top and bottom of the fraction by the root that's in the denominator. This clears the radical from the bottom without changing the value.
For √(1/2), first split it into √1/√2 = 1/√2. Multiply top and bottom by √2 to get √2/2. The calculator gives √2/2 for sqrt(1/2), and √6/3 for sqrt(2/3) using the same steps.
Why isn't √(a + b) equal to √a + √b?
Square roots split over multiplication, but they don't split over addition. So √(a × b) = √a × √b is true, while √(a + b) = √a + √b is false.
A quick test shows it. √(9 + 16) = √25 = 5, but √9 + √16 = 3 + 4 = 7. Since 5 isn't 7, you can only simplify a sum after you add the numbers inside.
Another frequent mistake is stopping too early. Writing √72 = √4 × √18 = 2√18 is a correct step, but 18 still contains the perfect square 9. Continuing gives 2 × 3√2 = 6√2. Using the largest perfect square from the start avoids the extra round.
How do you know a radical is fully simplified?
A square root is in simplest form when three things are true. The number inside has no perfect square factor other than 1. There's no fraction inside the root. No root is left in a denominator. Cube roots follow the same rules with perfect cubes.
Some roots can't be simplified at all. √17 stays √17 because 17 is prime. Others simplify to a whole number: √36 is just 6. This radical calculator handles all three cases and shows which factor it pulled out.
To use it, type the root the way you'd say it: sqrt(72) for the square root of 72, or cbrt(54) for the cube root of 54. A number written right before sqrt is treated as a coefficient, so 3sqrt(12) means 3√12. The steps list the factor pulled out, the simplified root and, for fractions, the step that rationalizes the denominator.
Which perfect squares should you check first?
Start with the largest perfect square that could fit and work down. Knowing the squares up to 100 by heart makes most textbook problems quick.
If 4 divides the number, try 16 and 36 too, since they're also multiples of 4. Many students find 4 first and then have to simplify again. That isn't wrong, but it takes longer: √48 = 2√12 = 2 × 2√3 = 4√3, while dividing out 16 gets there in one step.
| Radical | Largest perfect square factor | Simplified |
|---|---|---|
| √48 | 16 | 4√3 |
| √50 | 25 | 5√2 |
| √75 | 25 | 5√3 |
| √98 | 49 | 7√2 |
| √200 | 100 | 10√2 |
Where do you need to simplify radicals in algebra?
The most common place is the answer to a quadratic equation. Teachers usually expect the radical in simplest form, not a decimal.
Solving x² = 72 gives x = ±√72, which simplifies to x = ±6√2. The quadratic formula works the same way. For x² − 4x − 8 = 0 the discriminant is 48, and √48 = 4√3, so x = (4 ± 4√3)/2 = 2 ± 2√3. The quadratic equation solver on this site shows those steps in full.