Free · Powers and roots
Fifth Root and Integer-Order Real Roots
Find signed real roots, change the integer order, and distinguish negative odd roots from even-order inputs without a real result.
- Formula & worked example
- Private in your browser
- No signup
Calculator inputs
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How to use this calculator
- Enter the known values in the units shown. Results update as you type.
- Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
- Use the worked example to check the method. Reset restores the starting fields.
Private by default
Inputs and results stay in this browser tab. Bookify does not upload or store the values you enter.
Formula and method
The fifth root of −32 is −2 because (−2)^5=−32.
x^n=a; odd n permits negative a, even n selects the nonnegative root.
Worked example
Enter these known values and leave the other values blank.
- Radicand
- -32
- Integer root order
- 5
- Principal real root
- -2
Assumptions and limitations
- The root order starts at5 and remains editable. Use an integer greater than1 and below1e50; the upper limit is computational, not a mathematical restriction.
- Odd orders have one real root for every signed real radicand. Even orders require a nonnegative radicand and return the principal nonnegative root.
- The captured solver direction is radicand/order to result. This page does not promise an inverse root-order solver. Zero has root zero at every supported order.
- Very small nonzero values may appear in scientific notation when fixed decimal display cannot represent them. Their value is preserved rather than shown as zero.
Common questions
Does a negative number have a fifth root?
Yes. A negative number raised to an odd power remains negative, so the unique real fifth root is negative.
Why is a negative square root blank?
A real number squared cannot be negative. Choose a nonnegative radicand or an odd order; complex roots are outside this tool.
References
Translate the hash-pinned captured helper into equivalent declarative arithmetic without executing browser JavaScript. Use the signed real principal root for odd orders, and reject negative radicands at even orders instead of returning a hidden positive magnitude. All source order controls remain available. Evaluate the integer-order and signed-radicand conditions exactly by sign/modulo. Root orders must be integral, greater than one and below1e50 for reliable integer parity in the64-digit runtime. Negative inputs have real roots only at odd orders. When ordinary fixed-decimal rounding cannot show a finite nonzero value, preserve that value in scientific notation. This fallback is bound to the complete reviewed source and compiled schemas; ordinary rounding, equations and controls stay unchanged. The integer predicate is exact, including fractional orders arbitrarily close to a whole number.