Fifth Roots Calculator finds the fifth root of a number, returning an exact integer when the number is a perfect fifth power and a 12-digit decimal approximation otherwise. The fifth root of x is the number that, multiplied by itself five times, produces x.
Find the fifth root of a perfect fifth power
A perfect fifth power is a number that equals some integer raised to the fifth power. 3^5 = 243, so the fifth root of 243 is exactly 3. Fifth Roots Calculator checks this by searching for an integer whose fifth power matches the input exactly, and returns that integer directly whenever the match holds, with no rounding involved.
Work through the calculation for 243
To confirm the fifth root of 243 is 3 by hand, multiply 3 by itself five times: 3 × 3 = 9, 9 × 3 = 27, 27 × 3 = 81, 81 × 3 = 243. Five multiplications land exactly on 243, confirming the fifth root.
Fifth Roots Calculator shows this relationship as "243 is 3^5" rather than displaying a lengthy multiplication chain.
Approximate the fifth root of a non-perfect power
Most integers are not perfect fifth powers. The fifth root of 100 falls between the fifth root of 3^5 = 243 and the fifth root of 2^5 = 32, since 100 sits between those two values; the exact result is approximately 2.512.
Fifth Roots Calculator computes this using a fractional exponent internally, x^(1/5), and rounds the display to 12 significant digits.
Simplify a fifth root with a perfect-power factor
When a number is not a perfect fifth power outright but contains one as a factor, Fifth Roots Calculator can pull that factor out of the radical.
The fifth root of 96 factors as the fifth root of (32 × 3), and since 32 = 2^5, this simplifies to 2 times the fifth root of 3, rather than a single unsimplified decimal.
This mirrors how square roots pull out perfect-square factors, extended to the fifth-power case.
Handle negative inputs correctly
Fifth roots are odd-index roots, so negative inputs have a well-defined real fifth root, unlike square roots or fourth roots. The fifth root of −243 is −3, since (−3)^5 = −243 (an odd number of negative factors keeps the product negative).
Fifth Roots Calculator returns this negative result directly instead of treating negative inputs as an error, which is the correct behavior for any odd-index root.
Avoid this common mistake
Treating an odd root like an even root leads to skipped or incorrect negative answers. Even roots, such as square roots and fourth roots, are undefined for negative real inputs, but fifth roots, seventh roots, and other odd-index roots are always defined for negative numbers.
Do not apply the "no negative inputs" rule from square root problems to fifth root problems; the parity of the root index changes which inputs are valid.
Work through a second perfect-power example
The fifth root of 1,024 is 4, since 4^5 = 4 × 4 × 4 × 4 × 4 = 1,024. Building this up step by step: 4 × 4 = 16, 16 × 4 = 64, 64 × 4 = 256, 256 × 4 = 1,024, confirming the fifth power lands exactly on the target.
Fifth Roots Calculator recognizes 1,024 as a perfect fifth power immediately and returns the clean integer 4 rather than an unnecessary decimal approximation.
Relate fifth roots to fractional exponents
A fifth root is defined identically to raising a number to the power 1/5, so every rule that applies to exponents also applies to fifth roots.
Multiplying two fifth roots of the same structure, such as the fifth root of a times the fifth root of b, equals the fifth root of (a × b), following the same product rule that governs ordinary exponents.
This connection is why Fifth Roots Calculator can fall back on the general power function internally while still presenting the result as a root.
Frequently asked questions
What is the fifth root of 243?
The fifth root of 243 is 3, since 3 raised to the fifth power (3 × 3 × 3 × 3 × 3) equals 243 exactly.
How do you find a fifth root by hand?
To find a fifth root by hand for a suspected perfect fifth power, test small integers by raising each to the fifth power until one matches the target number. For non-perfect powers, use the fractional exponent x^(1/5) and a calculator for the decimal approximation.
Can a fifth root be negative?
A fifth root can be negative, since fifth roots are an odd-index root and every real number, positive or negative, has exactly one real fifth root. The fifth root of −32 is −2.
What is the difference between a fifth root and a square root?
A fifth root asks what number multiplied by itself five times gives the target, while a square root asks what number multiplied by itself twice gives the target. Fifth roots are odd-index and accept negative inputs; square roots are even-index and do not.
How do you simplify the fifth root of a non-perfect fifth power?
To simplify the fifth root of a non-perfect fifth power, factor out any perfect fifth-power factor from inside the radical and move it outside as a coefficient, leaving the remaining factor under the fifth root sign.
Is the fifth root the same as raising to the power of 1/5?
The fifth root is exactly the same operation as raising a number to the power of 1/5, since x^(1/5) is defined as the number that, raised to the fifth power, returns x.
What is the fifth root of 1,024?
The fifth root of 1,024 is 4, since 4 raised to the fifth power (4 × 4 × 4 × 4 × 4) equals 1,024 exactly.
Does the fifth root follow the same product rule as exponents?
The fifth root follows the same product rule as exponents, since it is defined as raising to the power 1/5; the fifth root of a times the fifth root of b equals the fifth root of the product a times b.
Summary
Fifth Roots Calculator computes the fifth root of a number, returning an exact integer for perfect fifth powers like 243 (root 3) and a 12-digit decimal approximation otherwise. Because the fifth root is an odd-index root, negative inputs are valid and return negative results, unlike square roots or fourth roots.
Enter any real number to see its exact or approximate fifth root with the perfect-power check shown.