Exponent Calculator
Calculate powers and exponents with step-by-step solutions
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📚 Examples, Rules & Help
⚡Quick Examples - Try These Calculations
🔍How It Works
Exponent Rules
Basic Rules:
• a^0 = 1 (zero exponent)
• a^1 = a (identity)
• a^(-n) = 1/a^n (negative exponent)
• a^(1/n) = ⁿ√a (fractional exponent)
• a^m × a^n = a^(m+n) (product rule)
• (a^m)^n = a^(mn) (power rule)
Special Cases
Important Cases:
• 0^n = 0 (n >= 0)
• 0^0 = undefined
• 1^n = 1 (any n)
• (-a)^n: positive if n is even, negative if n is odd
• Fractional exponents with negative bases may be complex
Applications
Common Uses:
• Scientific notation: 6.02 × 10^23
• Compound interest: A = P(1 + r)^t
• Population growth: P = P₀e^(rt)
• Computer science: 2^n complexity
• Physics: inverse square law ∝ 1/r²
🌍Real-World Applications
❓Frequently Asked Questions
What does a negative exponent mean?
A negative exponent means "take the reciprocal and make the exponent positive".
Example: 2⁻³ = 1/2³ = 1/8 = 0.125
It's a way to express division using exponent notation.
How do fractional exponents work?
Fractional exponents represent roots and powers combined.
a^(1/n) = nth root of a
a^(m/n) = nth root of a^m
Example: 8^(2/3) = ∛(8²) = ∛64 = 4
What happens with negative bases?
Integer exponents: Follow sign rules
• Even exponent: result is positive
• Odd exponent: result is negative
Fractional exponents: May result in complex numbers
Example: (-4)² = 16, but (-4)^0.5 is complex
Why is 0^0 undefined?
Different mathematical approaches give different results:
• From 0^n pattern: 0^0 should be 0
• From a^0 pattern: 0^0 should be 1
Since these conflict, mathematicians consider 0^0 undefined or indeterminate.
🎯Common Use Cases
🎓 Academic Learning
- • Algebra homework and exams
- • Scientific notation practice
- • Calculus preparation
- • Physics problem solving
💼 Professional Applications
- • Engineering calculations
- • Scientific research
- • Data analysis and modeling
- • Quality control measurements
💰 Financial Calculations
- • Compound interest calculations
- • Investment growth projections
- • Loan amortization
- • Economic modeling
💻 Technology & Computing
- • Algorithm complexity analysis
- • Binary and hexadecimal conversions
- • Database optimization
- • Network capacity planning