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Exponent Calculator

Calculate powers, roots, exponentials, and logarithms. Solve xⁿ, ⁿ√x, eˣ, and log expressions.

Exponent Worksheet

Exponent Operations

An exponent tells you how many times to multiply a number by itself. This calculator handles powers, roots, exponentials, and logarithms.

Power (xⁿ)

Multiply the base by itself n times.

  • 2⁴ = 2 × 2 × 2 × 2 = 16
  • 10³ = 10 × 10 × 10 = 1000
  • 5⁰ = 1 (any number to power 0 is 1)
  • 2⁻³ = 1/2³ = 1/8 = 0.125 (negative exponent = reciprocal)

Root (ⁿ√x)

The inverse of exponentiation. Find what number multiplied n times gives x.

  • √16 = 4 (because 4² = 16)
  • ∛27 = 3 (because 3³ = 27)
  • Fractional exponents are roots: x^(1/n) = ⁿ√x, so 8^(1/3) = ∛8 = 2

Exponential (eˣ)

e ≈ 2.71828 raised to the power x. Appears in compound interest, population growth, radioactive decay, and probability. e¹ = 2.718, e² = 7.389, e⁻¹ = 0.368.

Logarithm

The inverse of exponentiation. log₁₀(x) asks: "10 to what power equals x?"

  • log(100) = 2 (because 10² = 100)
  • log(1000) = 3 (because 10³ = 1000)
  • ln(x) = natural log, using base e instead of 10

Laws of Exponents

These rules simplify expressions involving exponents:

  • Product rule: aᵐ × aⁿ = aᵐ⁺ⁿ — multiply same base, add exponents. Example: 2³ × 2⁴ = 2⁷ = 128
  • Quotient rule: aᵐ ÷ aⁿ = aᵐ⁻ⁿ — divide same base, subtract exponents. Example: 5⁶ ÷ 5² = 5⁴ = 625
  • Power of a power: (aᵐ)ⁿ = aᵐˣⁿ — multiply exponents. Example: (2³)² = 2⁶ = 64
  • Power of a product: (ab)ⁿ = aⁿ × bⁿ — distribute exponent. Example: (3×4)² = 3² × 4² = 144
  • Zero exponent: a⁰ = 1 (for any non-zero a)
  • Negative exponent: a⁻ⁿ = 1/aⁿ — flip to a fraction

Real-World Applications

  • Compound interest: A = P(1 + r/n)^(nt) uses exponents to calculate investment growth
  • Scientific notation: 3.0 × 10⁸ m/s (speed of light) — exponents express very large or small numbers
  • Computer science: Binary storage uses powers of 2. A 10-bit number stores 2¹⁰ = 1,024 values
  • Population growth: Exponential growth P(t) = P₀ × eʳᵗ models bacterial and population growth

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Frequently Asked Questions

What is an exponent?

An exponent tells you how many times to multiply a number by itself. 2³ means 2 × 2 × 2 = 8.

What is a negative exponent?

A negative exponent means divide instead of multiply. 2⁻³ = 1/2³ = 1/8 = 0.125.

What is e (Euler's number)?

e ≈ 2.71828 is a mathematical constant used in exponential growth, compound interest, and calculus.

What is a logarithm?

Logarithm is the inverse of exponentiation. log base 10 of 100 = 2 because 10 to the power 2 = 100.

What does any number raised to the power of zero equal?

Any non-zero number raised to the power of zero equals 1. For example, 5 to the power 0 = 1 and 1000 to the power 0 = 1. The case of 0 to the power 0 is mathematically undefined.

How do you calculate 2 to the power of 10?

Multiply 2 by itself 10 times: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024. So 2 to the power 10 = 1024, a value commonly used in computing as 1 kibibyte.

What is the difference between log and ln?

log (base 10) asks what power of 10 gives the value: log(1000) = 3. ln is the natural log using base e (2.71828): ln(7.389) is approximately 2. Both are inverses of their respective exponential functions.

What is a fractional exponent?

A fractional exponent represents a root. x to the power 1/2 equals the square root of x, and x to the power 1/3 equals the cube root of x. For example, 27 to the power 1/3 = 3, because 3 times 3 times 3 = 27.