Rational Exponents Calculator
Calculate values with rational (fractional) exponents using step-by-step solutions
Calculate Rational Exponents
The number that will be raised to the rational exponent
Top part of the fraction
Bottom part of the fraction
Expression: 00/1
Result
Example Calculations
Example 1: 8^(1/3)
Base: 8
Exponent: 1/3
Calculation: ³√8 = 2
Result: 2
Example 2: 27^(2/3)
Base: 27
Exponent: 2/3
Calculation: (³√27)² = 3² = 9
Result: 9
Example 3: 16^(3/4)
Base: 16
Exponent: 3/4
Calculation: (⁴√16)³ = 2³ = 8
Result: 8
Example 4: 32^(2/5)
Base: 32
Exponent: 2/5
Calculation: (⁵√32)² = 2² = 4
Result: 4
Rational Exponent Rules
Basic Form
a^(m/n) = ⁿ√(a^m) = (ⁿ√a)^m
Square Root
a^(1/2) = √a
Cube Root
a^(1/3) = ³√a
Negative Exponent
a^(-m/n) = 1/a^(m/n)
Properties
The denominator indicates the type of root
The numerator indicates the power
Even roots of negative numbers are undefined
Can be calculated as root first, then power
Understanding Rational Exponents
What are Rational Exponents?
A rational exponent is an exponent that is expressed as a fraction. The notation a^(m/n) represents the nth root of a raised to the mth power, which can be calculated as either (ⁿ√a)^m or ⁿ√(a^m).
Key Components
- •Base (a): The number being raised to a power
- •Numerator (m): The power to which the root is raised
- •Denominator (n): The index of the root
Calculation Methods
Method 1: Root first, then power
a^(m/n) = (ⁿ√a)^m
Method 2: Power first, then root
a^(m/n) = ⁿ√(a^m)
Tip: Method 1 is often easier when dealing with smaller numbers, as it avoids calculating large powers first.
Common Applications
Geometry
Volume and surface area calculations involving fractional dimensions
Physics
Energy calculations and wave equations with fractional powers
Finance
Compound interest calculations with fractional compounding periods