How this instrument works
Just as a perfect-square factor can be pulled out from under a square root, a perfect-CUBE factor can be pulled out from under a cube root: ∛(a³b) equals a∛b exactly, for positive a and b. Both sides of that identity are computed here independently — the left directly from the un-simplified expression, the right from the already-simplified form — confirming they land on the same value.
This is the cube-root sibling of this site's square-root Radical calculator, using the identical extraction idea one root-index higher: instead of asking 'what's the largest perfect square hiding inside this number,' the question becomes 'what's the largest perfect cube hiding inside it,' and that cube's own cube root steps outside the radical entirely.
This page covers the positive-real case specifically — both a and b entered as positive numbers — since cube roots of negative numbers, unlike square roots, do have a well-defined real value, but folding that sign case in cleanly needs its own separate handling this page doesn't attempt.
- Enter the value to be cubed (the perfect-cube factor's own root) into the a field.
- Enter the remaining factor left under the cube root into the b field.
- Read ∛(a³b) — computed directly from the unsimplified expression.
- Read a∛b — computed via the already-simplified form, and confirm the two values match.
Worked example — simplifying ∛(2³×5)
With a=2 and b=5: computed directly, ∛(2³×5)=∛40≈3.420. Computed via the simplified form, 2∛5≈2×1.710=3.420 — both routes agree to the same value, confirming that pulling the perfect-cube factor (2³=8) outside the radical, leaving just its cube root (2) multiplied by the remaining factor (5)'s own cube root, changes nothing about the actual value.
With a=3 and b=2: ∛(3³×2)=∛54≈3.780, and 3∛2≈3×1.260=3.780 — the identity holds again with a larger extracted factor. With a=1 (no perfect-cube factor at all), both sides trivially reduce to ∛10≈2.154, since a=1 contributes nothing to extract.
Questions
What does it mean to simplify a cube root?
Pulling any perfect-cube factor out from underneath the radical sign, leaving only its cube root outside and the remaining factor underneath — exactly parallel to simplifying a square root by extracting a perfect-square factor instead.
How is this different from simplifying a square root?
The extracted factor must be a perfect CUBE (a³) rather than a perfect square (a²), and the entire operation is one root-index higher — otherwise the underlying idea, pulling a repeated factor outside the radical, is identical.
Why does this page only cover positive numbers?
Cube roots of negative numbers do have a well-defined real value (unlike square roots of negatives), but handling that sign case correctly needs separate logic this page doesn't implement — it's scoped here to the positive-real case only.
What if a is 1?
There's no perfect-cube factor to extract at all, so both sides of the identity simply reduce to the plain cube root of b — the identity still holds, just trivially, with nothing actually simplified.
Is this the same as this site's Radical calculator?
No — that page demonstrates the analogous identity for SQUARE roots (√(a²b)=a√b); this page demonstrates the CUBE-root version of the identical extraction idea, one root-index higher.