What is the rule for negative exponents?

A negative exponent represents the reciprocal of the base raised to the positive power: x^(-y) = 1 / (x^y). For example, 5^(-2) = 1 / 5^2 = 1 / 25 = 0.04.

How are fractional exponents evaluated?

A fractional exponent x^(a/b) represents the b-th root of x raised to the power a: x^(a/b) = (b√x)^a. For example, 27^(2/3) = (3√27)^2 = 3^2 = 9.

Why is any non-zero number raised to the zero power equal to 1?

By the quotient rule of exponents, x^a / x^a = x^(a - a) = x^0. Since any non-zero quantity divided by itself equals 1, x^0 must equal 1.

Exponent & Power Calculator - Laws of Exponents

The Universal Laws & Properties of Exponents

In algebra, exponentiation is a mathematical operation written as x^y, involving a base x and an exponent y. When y is a positive integer, exponentiation corresponds to repeated multiplication of the base.

Core Exponent Rules Reference Sheet

Exponent Rule Name	Algebraic Formula	Practical Example
Product Rule	x^a × x^b = x^{a + b}	2³ × 2⁴ = 2⁷ = 128
Quotient Rule	x^a / x^b = x^{a - b}	5⁶ / 5² = 5⁴ = 625
Power of a Power	(x^a)^b = x^{a × b}	(3²)³ = 3⁶ = 729
Negative Exponent Rule	x^-a = 1 / x^a	4⁻² = 1 / 4² = 1 / 16 = 0.0625
Fractional Exponent Rule	x^a/b = ^b√(x^a)	8^2/3 = (³√8)² = 2² = 4
Zero Exponent Rule	x⁰ = 1 (for x ≠ 0)	999⁰ = 1

Frequently Asked Questions (FAQ)

What does a negative base with an exponent mean?

Parentheses are critical: (-2)⁴ = (-2) × (-2) × (-2) × (-2) = +16, whereas -2⁴ = -(2 × 2 × 2 × 2) = -16.

Can an exponent be a decimal?

Yes. Any decimal can be written as a rational fraction: x^1.5 = x^(3/2) = √(x³). For example, 4^1.5 = √(4³) = √64 = 8.