Positive and Negative Exponents Calculator

Exponents are a fundamental concept in mathematics that represent repeated multiplication. This calculator helps you understand and compute both positive and negative exponents with ease.

What Are Exponents?

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

aⁿ means a multiplied by itself n times.

For example, 2³ means 2 × 2 × 2 = 8. Here, 2 is the base and 3 is the exponent.

Positive Exponents

Positive exponents represent repeated multiplication of the base. For any positive integer n:

aⁿ = a × a × ... × a (n times)

Examples:

3² = 3 × 3 = 9
5³ = 5 × 5 × 5 = 125

Negative Exponents

Negative exponents represent reciprocals of the base raised to the positive exponent. For any positive integer n:

a^-n = 1 / aⁿ

Examples:

2^-3 = 1 / 2³ = 1/8
4^-2 = 1 / 4² = 1/16

Exponent Rules

Product of Powers

a^m × aⁿ = a^m+n

Example: 2³ × 2⁴ = 2⁷ = 128

Quotient of Powers

a^m / aⁿ = a^m-n

Example: 5⁶ / 5² = 5⁴ = 625

Power of a Power

(a^m)ⁿ = a^m×n

Example: (3²)³ = 3⁶ = 729

Examples

Positive Exponent Example

Calculate 4³:

4 × 4 × 4 = 64

Negative Exponent Example

Calculate 7^-2:

1 / 7² = 1/49

FAQ

What is the difference between positive and negative exponents?: Positive exponents represent repeated multiplication, while negative exponents represent reciprocals of the base raised to the positive exponent.
Can exponents be zero?: Yes, any non-zero number raised to the power of zero is 1 (a⁰ = 1).
How do I simplify expressions with exponents?: Use exponent rules like product of powers, quotient of powers, and power of a power to simplify expressions.
What happens when I raise zero to a negative exponent?: Zero raised to a negative exponent is undefined (0^-n is undefined).
Can I use negative exponents in real-world calculations?: Yes, negative exponents are commonly used in scientific notation, physics, and engineering to represent very small numbers.