Fractional Exponents Calculator
Result:
Master Fractional Exponents with our comprehensive calculator. Understand rational exponents, convert between radical and exponential forms, and solve complex power calculations with ease.
What are Fractional Exponents?
Fractional exponents (rational exponents) are exponents written as fractions. x^(m/n) means the nth root of x raised to the mth power, or equivalently, x raised to the mth power then take the nth root.
Fractional Exponent Rules
x^(m/n) = ⁿ√(x^m) = (ⁿ√x)^m
Key Conversion Rules
x^(1/n) = ⁿ√x (nth root)
x^(m/n) = ⁿ√(x^m) (power then root)
x^(m/n) = (ⁿ√x)^m (root then power)
x^(-m/n) = 1/x^(m/n) (negative exponent)
Common Examples
8^(1/3) = ∛8 = 2
16^(3/4) = (⁴√16)³ = 2³ = 8
25^(1/2) = √25 = 5
27^(2/3) = (∛27)² = 3² = 9
Step-by-Step Calculation Method
Example: Calculate 32^(3/5)
Method 1: Root then Power
Step 1: Find the 5th root: ⁵√32 = 2
Step 2: Raise to the 3rd power: 2³ = 8
Result: 32^(3/5) = 8
Method 2: Power then Root
Step 1: Raise to the 3rd power: 32³ = 32,768
Step 2: Find the 5th root: ⁵√32,768 = 8
Result: 32^(3/5) = 8
Note: Method 1 is often easier when dealing with perfect roots!
Properties of Fractional Exponents
- Product rule: x^(m/n) × x^(p/q) = x^(m/n + p/q)
- Quotient rule: x^(m/n) ÷ x^(p/q) = x^(m/n - p/q)
- Power rule: (x^(m/n))^(p/q) = x^(mp/nq)
- Product of bases: (xy)^(m/n) = x^(m/n) × y^(m/n)
- Quotient of bases: (x/y)^(m/n) = x^(m/n) ÷ y^(m/n)
Special Cases and Important Notes
Unit Exponents
x^(1/n) = ⁿ√x
Example: 9^(1/2) = √9 = 3
Simplest fractional exponentNegative Exponents
x^(-m/n) = 1/x^(m/n)
Example: 8^(-1/3) = 1/2
Creates reciprocalsDomain Restrictions
Base must be positive
Why: Even roots of negatives
Avoids complex numbersReal-World Applications
- Finance: Compound interest calculations with fractional periods
- Physics: Scaling laws and dimensional analysis
- Engineering: Signal processing and power calculations
- Biology: Allometric scaling relationships
- Economics: Production function calculations
- Computer Science: Algorithm complexity analysis
Practice Problems
Basic Calculation
Calculate: 64^(2/3)
Answer: 16
(∛64)² = 4² = 16Negative Exponent
Calculate: 125^(-2/3)
Answer: 1/25
1/(∛125)² = 1/5² = 1/25Unit Fraction
Calculate: 729^(1/6)
Answer: 3
⁶√729 = 3Converting Between Forms
| Exponential Form | Radical Form | Decimal Exponent | Value |
|---|---|---|---|
| 16^(1/2) | √16 | 16^0.5 | 4 |
| 27^(2/3) | (∛27)² | 27^0.667 | 9 |
| 81^(3/4) | (⁴√81)³ | 81^0.75 | 27 |
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