acalculator

Need help simplifying radicals?

Type the number under the radical sign and the index. The calculator takes out every perfect square (or cube, or nth power) and shows the steps.

Your numbers

Simplest radical form
6√2

√72 in simplest radical form is 6√2.

Decimal value
8.485281
Number outside
6
Number inside
2
Radical
√72
Working
72 = 2^3 × 3^2; The largest perfect square factor is 36 = 6^2, so 72 = 36 × 2; √72 = 6√2

Simplest radical form: 6√2. √72 in simplest radical form is 6√2.

The working by hand

How to calculate

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.

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

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.

  • 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.

Machine-readable copies: Markdown, JSON.

Worked examples

Each example is checked against the calculator on every build.

  1. Number in front (a) 1, Index (n) 2, Radicand (x) 72 gives Simplest radical form 6√2, Number outside 6, Number inside 2, Decimal value 8.485281, Working 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. Number in front (a) 3, Index (n) 2, Radicand (x) 50 gives Simplest radical form 15√2, Number outside 15, Number inside 2, Radical 3√50.
  3. Number in front (a) 1, Index (n) 3, Radicand (x) -16 gives Simplest radical form -2∛2, Working ∛(-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. Number in front (a) 1, Index (n) 4, Radicand (x) 162 gives Simplest radical form 3∜2, Decimal value 3.567621.
  5. Number in front (a) 1, Index (n) 2, Radicand (x) 144 gives Simplest radical form 12, Number inside 1, Decimal value 12.
  6. Number in front (a) 2, Index (n) 5, Radicand (x) 96 gives Simplest radical form 4 × ⁵√3, Number outside 4, Number inside 3.
  7. Number in front (a) 1, Index (n) 2, Radicand (x) 30 gives Simplest radical form √30, Number outside 1, Number inside 30.

How it works

The calculator simplifies a × ⁿ√x, where a (the number in front) and x (the radicand) are whole numbers and n (the index) is a whole number from 2 to 10. It uses the product property of radicals, ⁿ√(k × m) = ⁿ√k × ⁿ√m, with exact whole-number arithmetic.

  1. Factor |x| into primes: |x| = p₁^e₁ × p₂^e₂ × … (smallest prime first).
  2. Each prime p with exponent e contributes p^⌊e ÷ n⌋ outside and p^(e mod n) inside. So |x| = kⁿ × m, where k = ∏ p^⌊e ÷ n⌋ and m = ∏ p^(e mod n). kⁿ is the largest perfect nth power that divides |x|, and m has no perfect nth-power factor other than 1.
  3. The result is c × ⁿ√m with c = a × k, and c negative when x is negative (only an odd n allows that, because an odd root keeps the sign: ∛−8 = −2).

An even root of a negative number is not a real number, so there is no answer. When x is 0 (or a is 0), the answer is 0.

The calculator shows:

  • Simplest radical form: c followed by the radical of m. The radical is written √m for n = 2, ∛m for n = 3, ∜m for n = 4, and with the index as superscript digits before √ for n from 5 to 10 (⁵√3). When c is 1 it is left out (√30), when c is −1 only a minus sign is written (-∛2), for n up to 4 c comes right before the radical (6√2), and for n of 5 or more c is followed by a space, × and a space (4 × ⁵√3), so the superscript index cannot be read as an exponent of c. When m is 1, or the answer is 0, only the whole number c is shown (12). A minus sign is a plain hyphen, and numbers have no thousands separators.
  • Decimal value: c × m^(1/n) (the same as a × ⁿ√x), shown to at most 6 decimal places.
  • Number outside: c. Number inside: m (1 for a whole-number answer, 0 when the answer is 0).
  • Radical: a × ⁿ√x as typed, written the same way, with a negative radicand in brackets (∛(-16)) a front number of 1 left out, and −1 written as a minus sign.
  • Working: the steps, joined by a semicolon and a space:
    1. For a negative x: ⁿ√(x) = -ⁿ√|x|, because an odd root keeps the sign (with the radical signs as above).
    2. For |x| of 2 or more: the prime factors, 72 = 2^3 × 3^2 (an exponent of 1 is not written).
    3. For |x| of 2 or more: The largest perfect square factor is 36 = 6^2, so 72 = 36 × 2 (square for n = 2, cube for n = 3, and 4th power, 5th power and so on for n of 4 or more), or 30 has no perfect square factor other than 1 when k is 1.
    4. The radical as typed, an equals sign with a space on each side, and the simplest form (√72 = 6√2).

Assumptions

  • a is from −1,000,000 to 1,000,000, x from −10^12 to 10^12, and n from 2 to 10.
  • The root is the principal real root: √x is never negative.

Worked examples by hand

√72. 72 = 2³ × 3². For a square root, each pair comes out: 2^⌊3÷2⌋ × 3^⌊2÷2⌋ = 2 × 3 = 6 outside, and 2^(3 mod 2) × 3^(2 mod 2) = 2 inside. The largest perfect square factor is 36 = 6², so √72 = 6√2 ≈ 8.485281.

3√50. 50 = 2 × 5². 5 comes out and 2 stays in, so √50 = 5√2 and 3√50 = 15√2.

∛(−16). An odd root keeps the sign: ∛(−16) = −∛16. 16 = 2⁴ = 2³ × 2, so ∛16 = 2∛2 and ∛(−16) = −2∛2 ≈ −2.519842.

∜162. 162 = 2 × 3⁴. 3 comes out and 2 stays in: ∜162 = 3∜2 ≈ 3.567621.

√144. 144 = 2⁴ × 3² = 12², so nothing stays inside: √144 = 12.

2 × ⁵√96. 96 = 2⁵ × 3. 2 comes out, so ⁵√96 = 2 × ⁵√3 and 2 × ⁵√96 = 4 × ⁵√3.

√30. 30 = 2 × 3 × 5 has no square factor other than 1, so √30 is already in simplest form.

Other questions people ask

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.