Exponent Calculator

Calculate a power

Enter a base and exponent to calculate the power result and see the inverse relationship.

What is an Exponent?

An exponent represents repeated multiplication of a base number. The expression x^y (read as "x to the power of y") means multiplying x by itself y times. For example, 2^3 = 2 × 2 × 2 = 8.

Exponents are fundamental in mathematics and science, used in scientific notation, compound interest calculations, exponential growth/decay models, and many areas of physics and engineering.

How Exponents Are Calculated

Follow these detailed steps:

  1. Step 1: Identify Base and Exponent
    In aⁿ, a is the base and n is the exponent (power). For 2³ = 8, base is 2, exponent is 3.
  2. Step 2: Apply Exponent Rules
    a^m × a^n = a^(m+n), (a^m)^n = a^(m×n), a^(-n) = 1/a^n, a^(1/n) = ⁿ√a.
  3. Step 3: Calculate the Result
    Multiply the base by itself n times. For scientific notation, 3.5 × 10⁶ = 3,500,000.

Formulas and Rules

x^n = x × x × x × ... × x (n times)

x^0 = 1 (any non-zero number)
x^1 = x
x^(-n) = 1 / x^n
x^(1/n) = n√x (nth root)
x^(m/n) = (n√x)^m

Exponent Laws:

  • x^a × x^b = x^(a+b)
  • x^a / x^b = x^(a-b)
  • (x^a)^b = x^(a×b)
  • (xy)^a = x^a × y^a
  • (x/y)^a = x^a / y^a

Examples

Calculate 2^5

Problem: Find 2 to the power of 5.

Solution:

  1. 2^5 = 2 × 2 × 2 × 2 × 2
  2. 2 × 2 = 4
  3. 4 × 2 = 8
  4. 8 × 2 = 16
  5. 16 × 2 = 32
  6. Result: 2^5 = 32

Calculate 10^(-2)

Problem: Find 10 to the power of -2.

Solution:

  1. 10^(-2) = 1 / 10^2
  2. 10^2 = 100
  3. Result: 10^(-2) = 0.01

Calculate 4^(1/2)

Problem: Find 4 to the power of 1/2.

Solution:

  1. 4^(1/2) = √4 (square root of 4)
  2. Result: 4^(1/2) = 2

Quick Calculation Tips

  • a⁰ = 1 for any non-zero a (including negative numbers)
  • Negative exponent means reciprocal: x^(-n) = 1/x^n
  • Fractional exponent is a root: x^(1/2) = √x, x^(1/3) = ³√x
  • 0⁰ is undefined (indeterminate form)

Common Mistakes to Avoid

  • (a + b)ⁿ ≠ aⁿ + bⁿ
    (x + 2)² = x² + 4x + 4, NOT x² + 4. Expand using distribution or binomial theorem.
  • Multiplying bases incorrectly
    2³ × 3² ≠ 6⁵. Different bases cannot be combined; 8 × 9 = 72.

Frequently Asked Questions

What is 0^0?

0^0 is considered indeterminate in most contexts, though in some fields like combinatorics, it's defined as 1. Most calculators will either return 1 or an error for this case.

How do negative exponents work?

A negative exponent means "take the reciprocal." For example, x^(-n) = 1/(x^n). So 2^(-3) = 1/(2^3) = 1/8 = 0.125.

What does a fractional exponent mean?

A fractional exponent represents a root. x^(1/n) means the nth root of x. For example, 8^(1/3) = ∛8 = 2. The numerator represents a power, so x^(m/n) = (n√x)^m.

How is this different from factorial?

Exponents involve repeated multiplication of the same base (x^y = x × x × ... × x), while factorial involves multiplying consecutive integers (n! = n × (n-1) × ... × 1).

Related Calculators