Negative and Positive Exponents Calculator

What Are Exponents?

An exponent indicates how many times a number (the base) is multiplied by itself. The general form is:

For example, 3⁴ means 3 multiplied by itself 4 times: 3 × 3 × 3 × 3 = 81.

Exponents can be positive, negative, zero, or fractions, each with specific rules and applications.

Positive Exponents

Positive exponents represent repeated multiplication of the base. The rules for positive exponents are straightforward:

• a¹ = a
• a² = a × a
• a³ = a × a × a
• And so on...

For example, 5³ = 5 × 5 × 5 = 125.

Negative Exponents

Negative exponents represent reciprocals of the base raised to the positive exponent. The key rule is:

For example, 2^-3 = 1 / 2³ = 1/8.

Negative exponents are particularly useful in algebra, calculus, and physics for representing very small quantities.

Key Exponent Rules

Here are the fundamental rules for working with exponents:

1. Product of Powers: a^m × aⁿ = a^m+n
2. Quotient of Powers: a^m / aⁿ = a^m-n
3. Power of a Power: (a^m)ⁿ = a^m×n
4. Negative Exponent: a^-n = 1 / aⁿ
5. Zero Exponent: a⁰ = 1 (for a ≠ 0)

These rules form the foundation for solving more complex exponent problems.

How to Use This Calculator

Our calculator makes it easy to compute both positive and negative exponents:

1. Enter the base number in the first field
2. Enter the exponent (positive or negative) in the second field
3. Click "Calculate" to see the result
4. View the step-by-step calculation in the result panel

The calculator handles all exponent rules automatically, including negative exponents which it converts to their reciprocal form.