How this instrument works
A rational exponent like x^(m⁄n) can be computed two different ways: raise x to the power m first, then take the nth root of that result — (xᵐ)^(1⁄n) — or take the nth root of x first, then raise that result to the power m — (x^(1⁄n))ᵐ. Both orders always agree, for any positive x, a genuine identity rather than a coincidence, and this calculator computes both routes side by side to make that agreement concrete rather than assumed.
The identity holds because a fractional exponent is fundamentally just a combination of an ordinary whole-number power and an ordinary root, and both of those operations commute with each other for a positive base — the order they're applied in doesn't change the final result, the same way 2×3 and 3×2 both give 6 regardless of which multiplication happens first.
This matters practically because it means whichever order is more convenient for a given number can be chosen freely: taking the root first often keeps the intermediate numbers smaller and easier to work with by hand (an 8th root of a modest number, then a small power, rather than a huge power computed first and an unwieldy root taken of that).
- Enter the base into the x field.
- Enter the power into the m field.
- Enter the root into the n field.
- Read both results: the sheet computes power-then-root and root-then-power independently, and they should agree.
Worked example — x=8, m=2, n=3
For x=8, m=2, n=3: computing (8²)^(1⁄3) — square 8 to get 64, then take the cube root — gives 4. Computing (8^(1⁄3))² instead — take the cube root of 8 to get 2, then square it — also gives 4. Both orders land on the identical answer, confirming x^(2⁄3) = 4 regardless of which operation happens first.
For x=27, m=2, n=3: (27²)^(1⁄3) and (27^(1⁄3))² both give 9 — the cube root of 27 is the clean whole number 3, making the root-first route noticeably easier to compute by hand than squaring 27 to 729 first and taking THAT number's cube root.
Questions
What does a rational exponent mean?
x^(m⁄n) combines an ordinary power and an ordinary root: raise x to the power m, then take the nth root of the result — or equivalently, take the nth root of x first, then raise that to the power m. Both orders give the same answer.
Why does the order not matter?
Because both power and root are operations that commute with each other for a positive base — applying them in either order produces the identical final result, the same way the order of two ordinary multiplications doesn't change their product.
Which order is easier to compute by hand?
Usually taking the root first, since it keeps the intermediate numbers smaller — an easy nth root followed by a small power is generally simpler than computing a large power first and then extracting a root from that much bigger number.
Does this identity hold for negative bases?
Not always in the real numbers — a negative base raised to a fractional power can require a complex number, since even roots of negative numbers aren't real. This calculator is scoped to positive bases, where the identity holds cleanly.
How is this related to the Root and Power of a Power pages?
Those pages compute a single root or a single power-of-a-power in one direction. This page instead verifies that combining both operations gives the same answer regardless of order — a check on the identity itself rather than a standalone evaluation.