Fifth Root Calculator

What Is the Fifth Root?

The fifth root of a number x is the value y that satisfies y⁵ = x. In radical notation it appears as ⁵√x, and in exponential notation as x^1/5 or equivalently x^0.2. This fifth root calculator applies that formula instantly for any real input, including negative numbers and decimals.

The Fifth Root Formula

The defining formula is:

⁵√x = x^1/5

This equivalence between radical notation and rational exponents is a core algebraic identity. Khan Academy explains that any nth root equals the radicand raised to the power 1/n: ⁿ√x = x^1/n. Substituting n = 5 yields the fifth root formula directly.

Deriving the Formula

The derivation rests on the inverse-operation principle. Define y = x^1/5. Raise both sides to the 5th power: y⁵ = (x^1/5)⁵ = x^(1/5)×5 = x¹ = x. This confirms that x^1/5 produces the value whose fifth power equals x. The ORCCA nth Root Properties reference formalizes this as the principal root property: for any real number x and positive integer n, (x^1/n)ⁿ = x.

Variables Explained

• x (the radicand): The number whose fifth root is computed. Fifth roots accept all real numbers because 5 is odd. A negative base raised to an odd exponent remains negative: (−2)⁵ = −32, so ⁵√(−32) = −2. This contrasts with square roots, which require non-negative inputs for real results.

Step-by-Step Calculation

1. Enter the radicand x into the input field.
2. The calculator applies x^1/5 using floating-point arithmetic.
3. The result y is displayed immediately; verify by computing y⁵ to confirm it equals the original x.
4. On a physical scientific calculator: type x, press the power key, enter 0.2, and press equals.

Worked Examples

• ⁵√32 = 2 (2⁵ = 32)
• ⁵√243 = 3 (3⁵ = 243)
• ⁵√100,000 = 10 (10⁵ = 100,000)
• ⁵√1,024 = 4 (4⁵ = 1,024)
• ⁵√50 ≈ 2.1867 (non-integer result, 50^0.2)
• ⁵√(−243) = −3 (odd-index root of a negative number)

Key Properties of Fifth Roots

The fifth root function exhibits several important mathematical properties that distinguish it from other roots. First, it is an odd function, meaning ⁵√(−x) = −⁵√x for all real x. Second, fifth roots are strictly increasing: if a < b, then ⁵√a < ⁵√b. Third, the function exhibits slower growth than linear for large positive x, since the exponent 0.2 is less than 1. These properties make fifth roots useful in scaling applications and comparative analysis across different magnitudes of data.

Real-World Applications

• Compound Annual Growth Rate (CAGR): The CAGR over exactly 5 years equals the fifth root of the ending-to-beginning value ratio, minus 1. An investment rising from $10,000 to $16,105 over 5 years has a CAGR of (16,105 ÷ 10,000)^1/5 − 1 = 10% per year.
• Geometric mean of five values: The geometric mean equals the fifth root of the product of all five values. The California Water Boards geometric mean guidance applies this to environmental datasets such as bacteria counts, where the arithmetic mean is distorted by outliers.
• Engineering scaling laws: Certain blast-wave propagation and material-scaling relationships involve fifth-power dependence, requiring fifth-root inversion to derive design parameters from measured outcomes.

Domain and Range

The fifth root function has domain (−∞, +∞) and range (−∞, +∞). It is a strictly increasing, odd function: ⁵√(−x) = −⁵√x. The function passes through (0, 0), (1, 1), and (−1, −1), and grows more slowly than a linear function for large x because the exponent 1/5 = 0.2 is less than 1. The West Texas A&M University Radicals tutorial provides additional worked examples and properties of nth root functions useful for further study.