Simplify Cube Root Calculator
Extract perfect cube factors and reduce any cube root to its simplest radical form.
∛ What is Simplifying a Cube Root?
Simplifying a cube root means rewriting ∛n in the form a∛b, where b contains no perfect cube factor greater than 1. You achieve this by factoring out the largest perfect cube that divides n. Take the cube root of that factor (which is an integer) as the coefficient outside the radical, and leave the remainder inside. The result is called the simplified radical form of the cube root.
For example, consider ∛72. Begin by finding all perfect cube factors of 72: since 72 = 8 × 9 and 8 = 2³, the number 8 is the largest perfect cube that divides 72. Extract it: ∛72 = ∛(8 × 9) = ∛8 × ∛9 = 2∛9. Now 9 = 3² has no perfect cube factor greater than 1, so 2∛9 is the fully simplified form. Both ∛72 and 2∛9 equal approximately 4.160168, but 2∛9 is the standard simplified representation used in algebra.
Simplification matters because it makes expressions easier to compare and combine. Adding ∛72 and ∛8 is cumbersome in unsimplified form, but in simplified form they become 2∛9 + 2, which is immediately readable. Simplified radical form is also required when submitting answers in algebra courses, standardized tests, and engineering calculations.
This calculator finds the largest perfect cube factor automatically. Simplify mode handles any integer between −1000 and 1000 and shows the full step-by-step factoring. Evaluate mode handles any real number (including decimals) and returns the decimal cube root to 8 places for numerical work.
📐 Formula
∛n = a∛b, where n = a³ × b
n = the original integer under the radical
a³ = the largest perfect cube factor of |n|
a = ∛(a³), the integer outside the radical (outside coefficient)
b = n ÷ a³, the remaining factor inside the radical (radicand)
Condition: b has no perfect cube factor greater than 1
Example: n = 72, a³ = 8 (largest perfect cube factor), a = 2, b = 72 ÷ 8 = 9
Result: ∛72 = 2∛9
📖 How to Use This Calculator
Steps
1
Enter an integer: type any whole number in the input field or drag the slider. The slider covers −1000 to 1000. For numbers outside that range, type directly in the field.
2
Click Calculate: the calculator finds the largest perfect cube factor of your input, extracts its cube root, and displays the simplified form a∛b.
3
Read the step explanation: the Step row below the grid shows the full factoring chain — for example ∛72 = ∛(8 × 9) = 2∛9 — so you can reproduce the method by hand.
4
Verify using the decimal: the Decimal Value row confirms the numerical result. Use this to double-check that the simplified form matches the original.
5
Evaluate mode for decimals: switch to Evaluate mode when you need the cube root of a non-integer or when you only need the decimal value, not the radical form.
💡 Example Calculations
Example 1: Simplify ∛72
Extracting the perfect cube factor 8
1
n = 72. Perfect cube factors of 72: check 1, 8, 27, 64. 8 divides 72 (72 ÷ 8 = 9). 27 does not (72 ÷ 27 is not an integer). 64 does not. Largest is 8.
2
a³ = 8, so a = 2. b = 72 ÷ 8 = 9. Check: 9 has no perfect cube factor > 1 (9 = 3², not divisible by 8 or 27).
3
Result: ∛72 = 2∛9 ≈ 4.160168.
∛72 = 2∛9
Try this example →Example 2: Simplify ∛500
Larger number with factor 125
1
n = 500. Perfect cubes up to 500: 1, 8, 27, 64, 125, 216, 343. Check which divide 500: 125 divides (500 ÷ 125 = 4). 216 and 343 do not. Largest is 125.
2
a³ = 125, so a = 5. b = 500 ÷ 125 = 4. Check: 4 has no perfect cube factor > 1.
3
Result: ∛500 = 5∛4 ≈ 7.937005.
∛500 = 5∛4
Try this example →Example 3: Simplify ∛(−216)
Negative perfect cube
1
n = −216. Take absolute value: 216 = 6³. Largest perfect cube factor is 216 itself.
2
a³ = 216, a = 6. b = 216 ÷ 216 = 1. Since b = 1, the cube root is a whole number.
3
Apply sign: ∛(−216) = −6 (a perfect cube, fully simplified).
∛(−216) = −6
Try this example →Example 4: Simplify ∛50 (no perfect cube factor)
Already in simplest form
1
n = 50. Perfect cube factors of 50: check 8 (50 ÷ 8 = 6.25, no), 27 (50 ÷ 27 ≈ 1.85, no). Only 1 divides 50 as a perfect cube.
2
a = 1, b = 50. Since a = 1 and b = 50, the cube root is already in simplest form: ∛50.
3
Decimal: ∛50 ≈ 3.684031.
∛50 = ∛50 (no simplification possible)
Try this example →❓ Frequently Asked Questions
What does it mean to simplify a cube root?
Simplifying a cube root means writing ∛n as a∛b where b contains no perfect cube factor greater than 1. You factor out the largest perfect cube that divides n, take its cube root as the coefficient outside the radical, and leave the rest inside. The result is mathematically equivalent but in the standard simplified form used in algebra.
How do you find the largest perfect cube factor?
List the perfect cubes in order: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000. Check each one, starting from the largest that could divide your number, and pick the biggest that divides it evenly. For 72, the candidates are 1 and 8 (27 and above are too large or do not divide evenly), so 8 is the largest perfect cube factor.
Is ∛72 and 2∛9 the same value?
Yes, they are identical. ∛72 equals 2∛9 exactly because 72 = 8 × 9 = 2³ × 9, and by the product rule of radicals ∛(2³ × 9) = ∛(2³) × ∛9 = 2 × ∛9. Both evaluate to approximately 4.160168. Simplification does not change the value; it only changes the form in which it is written.
How do you simplify cube roots of negative numbers?
Apply the rule ∛(−n) = −∛n. Simplify the positive version first, then add the minus sign. For ∛(−72): simplify ∛72 = 2∛9, then put the minus back: ∛(−72) = −2∛9. This works because the cube root of a negative number is always a real negative number, and (−a)³ = −(a³).
What if the number has no perfect cube factor other than 1?
The cube root is already in simplest radical form. Numbers like 2, 3, 4, 5, 6, 7, 9, 10, 11, and 50 have no perfect cube factor greater than 1, so ∛2, ∛3, ∛50, and similar expressions cannot be simplified further. The calculator will confirm this and show the decimal value.
What is the outside coefficient?
The outside coefficient is the integer a in a∛b. It equals the cube root of the largest perfect cube factor of n. For ∛72, the largest perfect cube factor is 8 = 2³, so the outside coefficient is ∛8 = 2. The outside coefficient is always a positive integer (or negative if the original number was negative).
How do you verify a simplified cube root?
Cube the outside coefficient and multiply by the inside radicand. The result should equal the original number. For 2∛9: outside coefficient cubed = 2³ = 8; multiply by inside radicand: 8 × 9 = 72. That matches the original, so 2∛9 is the correct simplification of ∛72.
How do you add or subtract simplified cube roots?
You can only combine cube root terms that have the same radicand. 2∛9 + 5∛9 = 7∛9 because both terms have ∛9. But 2∛9 + 3∛2 cannot be combined because the radicands 9 and 2 differ. Always simplify each cube root first so like radicands become visible — for example ∛72 + ∛144 = 2∛9 + 2∛18, and since ∛9 ≠ ∛18 these cannot be combined further.
Does the calculator work for numbers larger than 1000?
Yes. The slider only covers −1000 to 1000 for convenience, but you can type any integer directly into the input field. For example, typing 1728 gives ∛1728 = 12 (since 1728 = 12³), and typing 5000 gives 10∛5 because 5000 = 1000 × 5 and 1000 = 10³.
Why is simplified radical form preferred in mathematics?
Simplified radical form is a canonical (unique, standard) representation. It makes it easy to compare expressions: seeing 2∛9 and 2∛9 immediately shows they are equal, whereas ∛72 and ∛72 are obviously equal too, but ∛72 and 2∛9 look different and require simplification to reveal equivalence. Simplified form also reduces errors when combining radicals algebraically.
What is the difference between the Simplify and Evaluate modes?
Simplify mode accepts integers, finds the exact simplified radical form a∛b, and keeps the result exact. Evaluate mode accepts any real number (including decimals) and returns the cube root as an 8-decimal approximation. Use Simplify when you need an exact algebraic answer for homework or equations; use Evaluate when you need a numerical value for measurements or computations.