Exponent Calculator
Raise any number to any power. Supports whole numbers, fractions, decimals, and negative exponents.
Exponent Rules
Understanding exponent rules makes complex calculations simple. Here are the key rules that govern how exponents work:
xⁿ x xᵐ = xⁿ⁺ᵐ (product rule)
xⁿ / xᵐ = xⁿ⁻ᵐ (quotient rule)
(xⁿ)ᵐ = xⁿ×ᵐ (power rule)
x⁰ = 1 (zero exponent, x ≠ 0)
x⁻ⁿ = 1 / xⁿ (negative exponent)
Fractional Exponents
A fractional exponent like x^(1/2) means a root. x^(1/2) is the square root of x. x^(1/3) is the cube root. x^(2/3) means the cube root of x, squared. In general, x^(a/b) = the b-th root of x, raised to the a-th power. This is how exponents and roots are connected.
Negative Exponents
A negative exponent means "take the reciprocal." 2^(-3) = 1 / 2³ = 1/8 = 0.125. This is useful in scientific notation for very small numbers: 5 x 10^(-4) = 0.0005. Negative exponents do not make the result negative; they make it a fraction.
What is 0 raised to the 0?
This is debated in mathematics. Most conventions define 0^0 as 1 because it simplifies many formulas in combinatorics and calculus. However, some contexts leave it undefined. This calculator returns 1, which matches most math software and textbook conventions.
Can you raise a negative number to a fractional power?
Only if the denominator of the fraction is odd. (-8)^(1/3) = -2 (cube root of -8). But (-4)^(1/2) is undefined in real numbers because no real number squared gives -4. This calculator will show "Undefined" for these cases.
Why does any number to the 0 equal 1?
Think of it as a pattern: 2^3 = 8, 2^2 = 4, 2^1 = 2, so each step divides by 2. The next step would be 2^0 = 1. You can also use the quotient rule: x^n / x^n = x^(n-n) = x^0, and any number divided by itself is 1.