Exponent Calculator
Calculate base raised to the power of exponent
Operations
Power (xʸ)
Multiplies the base by itself y times.
Example: 2³ = 2 × 2 × 2 = 8
Root (ʸ√x)
Finds the number that when raised to y equals x.
Example: ³√27 = 3 (because 3³ = 27)
Logarithm (logₓy)
Finds the power x must be raised to get y.
Example: log₂8 = 3 (because 2³ = 8)
Understanding Exponents
Exponents (also called powers or indices) represent repeated multiplication. The expression xn means multiplying x by itself n times.
Exponent Rules
| Rule | Formula | Example |
|---|---|---|
| Product Rule | xᵃ × xᵇ = xᵃ⁺ᵇ | 2³ × 2² = 2⁵ = 32 |
| Quotient Rule | xᵃ ÷ xᵇ = xᵃ⁻ᵇ | 2⁵ ÷ 2² = 2³ = 8 |
| Power Rule | (xᵃ)ᵇ = xᵃˣᵇ | (2³)² = 2⁶ = 64 |
| Zero Exponent | x⁰ = 1 | 5⁰ = 1 |
| Negative Exponent | x⁻ⁿ = 1/xⁿ | 2⁻³ = 1/8 |
| Fractional Exponent | x^(1/n) = ⁿ√x | 8^(1/3) = ³√8 = 2 |
Logarithm Rules
| Rule | Formula |
|---|---|
| Product Rule | log(xy) = log(x) + log(y) |
| Quotient Rule | log(x/y) = log(x) - log(y) |
| Power Rule | log(xⁿ) = n × log(x) |
| Change of Base | logₐ(x) = log(x) / log(a) |
| Identity | logₐ(a) = 1, logₐ(1) = 0 |
Common Logarithms
- Common log (log₁₀): Written as "log" - used in science and engineering
- Natural log (logₑ or ln): Base e ≈ 2.718 - used in calculus and growth models
- Binary log (log₂): Base 2 - used in computer science
Applications
- Compound Interest: A = P(1 + r)ⁿ
- Population Growth: P = P₀eʳᵗ
- Radioactive Decay: N = N₀e⁻ᵏᵗ
- Sound Intensity (Decibels): dB = 10 log(I/I₀)
- pH Scale: pH = -log[H⁺]
- Earthquake Magnitude: Richter scale is logarithmic
Related Calculators
Frequently Asked Questions
What does a negative exponent mean?
A negative exponent means 'take the reciprocal.' x^(-n) = 1/(x^n). For example, 2^(-3) = 1/(2^3) = 1/8 = 0.125. Think of it as moving the base to the other side of a fraction line.
Why is anything to the power of 0 equal to 1?
This follows from the quotient rule: x^a ÷ x^a = x^(a-a) = x^0. Since any number divided by itself equals 1, x^0 must equal 1. This is true for all non-zero numbers. Note: 0^0 is typically considered undefined or 1 depending on context.
How do fractional exponents relate to roots?
A fractional exponent x^(1/n) equals the nth root of x. For example, 8^(1/3) = cube root of 8 = 2. More generally, x^(m/n) = (nth root of x)^m. So 8^(2/3) = (cube root of 8)^2 = 2^2 = 4.
What are the most important exponent rules to remember?
The key rules are: x^a × x^b = x^(a+b) (product rule), x^a ÷ x^b = x^(a-b) (quotient rule), and (x^a)^b = x^(ab) (power rule). These let you simplify most exponent expressions without calculating the actual values.