Exponent Calculator

Calculate powers, roots, and exponential functions with step-by-step solutions. Perfect for algebra, calculus, and scientific calculations.

Calculate Exponents

Operation Type

Base (a)

Exponent (b)

Formula:

Result = a^b

Display Settings

Decimal Places

Show calculation steps

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Quick Examples

Results

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Enter your values to calculate exponents, roots, and exponential functions.

How Exponents Work

Basic Exponent Rules

Product Rule:

a^m × a^n = a^(m+n)

Quotient Rule:

a^m ÷ a^n = a^(m-n)

Power Rule:

(a^m)^n = a^(m×n)

Zero Exponent:

a^0 = 1 (where a ≠ 0)

Special Cases

Negative Exponent:

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

Fractional Exponent:

a^(m/n) = ⁿ√(a^m)

Square Root:

√a = a^(1/2)

Cube Root:

³√a = a^(1/3)

Types of Exponential Calculations

Powers & Exponents

Integer Powers:

2³ = 8, 5² = 25

Decimal Powers:

2^2.5 = 5.657, 10^1.5 = 31.623

Negative Powers:

2^(-3) = 0.125, 5^(-2) = 0.04

Roots & Radicals

Square Roots:

√16 = 4, √25 = 5

Cube Roots:

³√27 = 3, ³√64 = 4

nth Roots:

⁴√16 = 2, ⁵√32 = 2

Exponential & Logarithmic

Natural Exponential:

e^2 ≈ 7.389, e^(-1) ≈ 0.368

Common Logarithm:

log₁₀(100) = 2, log₁₀(1000) = 3

Natural Logarithm:

ln(e) = 1, ln(e²) = 2

Example Calculations

Compound Interest Growth

Calculate the growth of $1000 invested at 5% annual interest for 10 years.

Amount = 1000 × (1.05)^10 = 1000 × 1.629 = $1,628.89

Population Growth

A bacteria population doubles every hour. Starting with 100 bacteria, how many after 5 hours?

Population = 100 × 2^5 = 100 × 32 = 3,200 bacteria

Radioactive Decay

Calculate the remaining amount of a substance with half-life of 3 years after 9 years.

Remaining = Initial × (1/2)^(9/3) = Initial × (1/2)^3 = Initial × 0.125

Frequently Asked Questions

What's the difference between exponents and logarithms?

Exponents and logarithms are inverse operations. If a^b = c, then log_a(c) = b. Exponents show repeated multiplication, while logarithms find the power needed to get a result.

How do I calculate fractional exponents?

Fractional exponents represent roots. For example, a^(1/2) = √a, and a^(m/n) = ⁿ√(a^m). You can also convert to decimal form: a^(3/4) = a^0.75.

What happens with negative bases and fractional exponents?

Negative bases with fractional exponents can result in complex numbers. For real results, the denominator of the fraction should be odd, or use absolute value.

Why is anything to the power of 0 equal to 1?

This follows from the quotient rule: a^m ÷ a^m = a^(m-m) = a^0. Since any number divided by itself equals 1, a^0 = 1 (except when a = 0).