# Need help simplifying radicals?

Writes a square root, cube root, or any nth root of a whole number in simplest radical form, such as √72 = 6√2, with the prime factors and the steps.

- Page: https://www.acalculator.org/math/simplifying-radicals-calculator
- JSON spec: https://www.acalculator.org/math/simplifying-radicals-calculator.json
- Version: 9f06f106ab39

## Default answer

Example with the default inputs (Number in front (a) 1, Index (n) 2, Radicand (x) 72): √72 in simplest radical form is 6√2.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| a | Number in front (a) | The whole number that multiplies the radical, as in 3√50; 1 when there is none. |
| n | Index (n) | Which root: 2 for a square root, 3 for a cube root, up to 10. |
| x | Radicand (x) | The whole number under the radical sign, up to 1 trillion in size. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| simplified | Simplest radical form | The radical with every perfect nth-power factor taken out, such as 6√2. |
| decimal | Decimal value | The value of the radical as a decimal. |
| outside | Number outside | The whole number in front of the simplified radical. |
| inside | Number inside | The radicand left under the radical sign; 1 when the root is a whole number. |
| radical | Radical | The radical as typed, such as 3√50. |
| steps | Working | The prime factors of the radicand, the largest perfect nth-power factor, and the simplified form. |

## Method

Factor the radicand into primes, x = kⁿ × m, where kⁿ is its largest perfect nth-power factor; then a × ⁿ√x = (a × k) × ⁿ√m.

## Assumptions

- The number in front and the radicand are whole numbers; the index is a whole number from 2 to 10.
- An odd root of a negative number is negative (∛−8 = −2); an even root of a negative number has no real value.
- The root is the principal root: √x is never negative.

## Worked examples

1. a = 1, n = 2, x = 72 gives simplified = 6√2, outside = 6, inside = 2, decimal = 8.485281, steps = 72 = 2^3 × 3^2; The largest perfect square factor is 36 = 6^2, so 72 = 36 × 2; √72 = 6√2. Source: Method from OpenStax Elementary Algebra 2e, section 9.2: https://openstax.org/books/elementary-algebra-2e/pages/9-2-simplify-square-roots.
2. a = 3, n = 2, x = 50 gives simplified = 15√2, outside = 15, inside = 2, radical = 3√50.
3. a = 1, n = 3, x = -16 gives simplified = -2∛2, steps = ∛(-16) = -∛16, because an odd root keeps the sign; 16 = 2^4; The largest perfect cube factor is 8 = 2^3, so 16 = 8 × 2; ∛(-16) = -2∛2.
4. a = 1, n = 4, x = 162 gives simplified = 3∜2, decimal = 3.567621.
5. a = 1, n = 2, x = 144 gives simplified = 12, inside = 1, decimal = 12.
6. a = 2, n = 5, x = 96 gives simplified = 4 × ⁵√3, outside = 4, inside = 3.
7. a = 1, n = 2, x = 30 gives simplified = √30, outside = 1, inside = 30.

## FAQ

### How do I simplify a square root?

Find the largest perfect square that divides the number, and take its square root out in front. 72 = 36 × 2, and √36 = 6, so √72 = √36 × √2 = 6√2.

### What is simplest radical form?

A radical is in simplest form when the number under the radical sign (the radicand) has no factor that is a perfect nth power other than 1. 6√2 is in simplest form; √72 is not, because 72 has the square factor 36.

### How do I simplify a cube root?

Look for perfect cube factors (8, 27, 64, 125, …). 16 = 8 × 2, so ∛16 = ∛8 × ∛2 = 2∛2. For a fourth root, look for fourth powers (16, 81, 256, …): ∜162 = ∜81 × ∜2 = 3∜2.

### How do prime factors help simplify radicals?

Write the radicand as a product of primes, then group equal primes in sets of n (the index). Each full set comes out as one factor. 72 = 2³ × 3² has one pair of 2s and one pair of 3s, so 2 × 3 = 6 comes out and one 2 stays inside: 6√2.

### What if there is a number in front of the radical?

Multiply it by what comes out. 3√50: 50 = 25 × 2, so √50 = 5√2, and 3 × 5√2 = 15√2.

### Can I take the root of a negative number?

An odd root of a negative number is negative, because (−2)³ = −8: ∛(−16) = −2∛2. An even root (square root, fourth root) of a negative number is not a real number, so the calculator gives no answer.

### Why is the answer sometimes a whole number?

When the radicand is itself a perfect nth power, nothing is left inside: √144 = 12 and ∛125 = 5.

## Sources

- OpenStax, Elementary Algebra 2e, section 9.2 Simplify Square Roots (product property of square roots): https://openstax.org/books/elementary-algebra-2e/pages/9-2-simplify-square-roots
- OpenStax, Intermediate Algebra 2e, section 8.2 Simplify Radical Expressions (higher roots): https://openstax.org/books/intermediate-algebra-2e/pages/8-2-simplify-radical-expressions