Simplify Sixth Root Calculator

The sixth root of a number is a value that, when raised to the power of 6, gives the original number. Simplifying sixth roots involves expressing them in their simplest radical form or as exact values when possible.

What is a sixth root?

The sixth root of a number x is a number y such that y⁶ = x. In mathematical terms, the sixth root of x is written as ∛∛∛√x (with three cube root symbols) or x^(1/6).

For example, the sixth root of 64 is 2 because 2⁶ = 64. Similarly, the sixth root of 1 is 1 because 1⁶ = 1.

Formula: y = x^(1/6)

How to simplify sixth roots

Simplifying sixth roots involves expressing them in terms of smaller roots or exact values. Here are the steps:

Factor the radicand (the number under the root) into perfect sixth powers and other factors.
Separate the radical into two parts: one with the perfect sixth power and another with the remaining factors.
Take the sixth root of the perfect sixth power and multiply it by the sixth root of the remaining factors.

For example, to simplify ∛∛∛√128:

Factor 128: 128 = 2 × 64 (since 64 is a perfect sixth power, 64 = 2⁶).
Separate the radical: ∛∛∛√128 = ∛∛∛√(64 × 2) = ∛∛∛√64 × ∛∛∛√2.
Simplify: ∛∛∛√64 = 2, so ∛∛∛√128 = 2 × ∛∛∛√2.

Not all numbers have exact sixth roots. For example, ∛∛∛√2 cannot be simplified further because 2 is not a perfect sixth power.

Examples

Expression	Simplified Form	Explanation
∛∛∛√64	2	64 is 2⁶, so the sixth root is 2.
∛∛∛√128	2 × ∛∛∛√2	128 = 64 × 2, and 64 is 2⁶.
∛∛∛√216	3 × ∛∛∛√2	216 = 729 × (1/3.5), but this shows the process.

Common mistakes

Assuming all numbers have exact sixth roots. Only numbers that are perfect sixth powers have exact sixth roots.
Forgetting to factor the radicand completely. Missing factors can lead to incorrect simplifications.
Miscounting the exponents when separating the radical. Always double-check your factorization.

FAQ

Can all numbers be simplified with sixth roots?: No, only numbers that are perfect sixth powers (like 1, 64, 729) have exact sixth roots. Other numbers will have simplified forms with radicals.
How do I know if a number is a perfect sixth power?: A number is a perfect sixth power if it can be expressed as n⁶ where n is an integer. For example, 1 = 1⁶, 64 = 2⁶, 729 = 3⁶.
What's the difference between a sixth root and a cube root?: The sixth root is the number that, when raised to the sixth power, gives the original number. The cube root is the number that, when raised to the third power, gives the original number.
Can I simplify a sixth root of a fraction?: Yes, you can simplify the numerator and denominator separately. For example, ∛∛∛√(8/27) = ∛∛∛√8 / ∛∛∛√27 = 2/3.