Fractional Exponent Calculator

Calculate am/n = n√(am) with step-by-step conversion and detailed solutions.

Enter Base and Exponent

8^(2/3)

Fractional Exponent (m/n)

Result

Value of 82/3
4
8^(2/3) = 4
Expression 8^(2/3)
As Radical 3√(8²)
Base Raised to m 8² = 64
nth Root of Result 3√64 = 4
Decimal Value 4

Step-by-Step Solution

a^(m/n) = n√(a^m) = (n√a)^m

Understanding Fractional Exponents

A fractional exponent (also called a rational exponent) is an exponent that is a fraction. The expression am/n means "raise a to the power m, then take the nth root" or equivalently "take the nth root of a, then raise to the power m." Both approaches give the same result.

The general formula is:

am/n = n√(am) = (n√a)m

Two Equivalent Methods

Method 1: Power First

Raise the base to the numerator power, then take the nth root.

a^(m/n) = n√(a^m)

Method 2: Root First

Take the nth root of the base, then raise to the numerator power.

a^(m/n) = (n√a)^m

Special Case: 1/n

When m = 1, the fractional exponent is simply the nth root.

a^(1/n) = n√a

Negative Exponent

A negative fractional exponent means take the reciprocal.

a^(-m/n) = 1 / a^(m/n)

Common Examples

  • 82/3 = 3√(82) = 3√64 = 4
  • 271/3 = 3√27 = 3
  • 163/4 = 4√(163) = 4√4096 = 8
  • 251/2 = √25 = 5
  • 4-1/2 = 1/√4 = 1/2 = 0.5

Why Are Fractional Exponents Useful?

Fractional exponents provide a unified notation for powers and roots. They follow the same exponent rules as integer exponents, making algebraic manipulation much easier. For example, x1/2 · x1/3 = x5/6 using the rule am · an = am+n.

Rules for Fractional Exponents

  • Product Rule: am/n · ap/q = a(m/n + p/q)
  • Quotient Rule: am/n ÷ ap/q = a(m/n - p/q)
  • Power Rule: (am/n)p/q = a(mp)/(nq)
  • Zero Exponent: a0 = 1 (for a ≠ 0)