Fifth Root Calculator (⁵√x)

Calculate the fifth root of any number with precision. This calculator handles both positive and negative numbers, identifies perfect fifth roots, and provides detailed step-by-step solutions.

Perfect Fifth Roots:

• 1 → 1

• 32 → 2

• 243 → 3

• 1,024 → 4

• 3,125 → 5

• 7,776 → 6

• 16,807 → 7

• 32,768 → 8

• 59,049 → 9

• 100,000 → 10

What is a Fifth Root?

The fifth root of a number is a value that, when raised to the fifth power, equals the original number. For example, 2 is the fifth root of 32 because 2⁵ = 32.

If ⁵√x = n, then n⁵ = x

⁵√32 = 2 because 2⁵ = 32

⁵√243 = 3 because 3⁵ = 243

Example Calculations

Perfect Fifth Roots

Small Numbers

  • ⁵√1 = 1 (since 1⁵ = 1)
  • ⁵√32 = 2 (since 2⁵ = 32)
  • ⁵√243 = 3 (since 3⁵ = 243)
  • ⁵√1,024 = 4 (since 4⁵ = 1,024)
  • ⁵√3,125 = 5 (since 5⁵ = 3,125)

Larger Numbers

  • ⁵√7,776 = 6 (since 6⁵ = 7,776)
  • ⁵√16,807 = 7 (since 7⁵ = 16,807)
  • ⁵√32,768 = 8 (since 8⁵ = 32,768)
  • ⁵√59,049 = 9 (since 9⁵ = 59,049)
  • ⁵√100,000 = 10 (since 10⁵ = 100,000)

Real-World Applications

Engineering

  • Material stress analysis
  • Fluid dynamics calculations
  • Structural design optimization
  • Vibration analysis

Physics

  • Quantum mechanics
  • Wave propagation
  • Particle physics
  • Statistical mechanics

Non-Perfect Fifth Roots

Irrational Results

⁵√100 ≈ 2.511886431509580

Verify:

(2.511886431509580)⁵ = 100

Properties of Fifth Roots

  • Every positive real number has exactly one real fifth root
  • Negative numbers have one real fifth root
  • The fifth root of a negative number is negative
  • Perfect fifth powers have rational fifth roots
  • Most fifth roots result in irrational numbers
  • The fifth root function is strictly increasing