Exponent Calculator
Powers & Expanded Form
Understanding Exponents
An exponent tells you how many times to multiply a base number by itself. It's a compact way of writing repeated multiplication, and the rules extend naturally to negative and fractional exponents too.
\(2^4 = 2 \times 2 \times 2 \times 2 = 16\) — the exponent simply counts how many times the base appears in the multiplication.
\(2^{-3} = \frac{1}{2^3} = \frac{1}{8} = 0.125\) — a negative exponent flips the result into a fraction, without changing the sign of the base.
Exponent Rules
Key Takeaways
- The expanded form only appears for small integer exponents. Writing out 50 multiplications wouldn't be useful, so it's shown for exponents between -10 and 10.
- Fractional exponents are roots in disguise. Use the Root Calculator when you specifically want to work with roots.
- Zero to a negative power is always undefined, since it requires dividing by zero.
Frequently Asked Questions
- Enter the base number.
- Enter the exponent (positive, negative, or decimal).
- Click "Calculate" to see the result and, for small whole-number exponents, its expanded multiplication form.
A negative exponent means "take the reciprocal." b^(-n) = 1 / b^n. For example, 2^(-3) = 1 / (2 × 2 × 2) = 1/8 = 0.125.
A fractional exponent represents a root. b^(1/2) is the square root of b, and b^(1/3) is the cube root. More generally, b^(m/n) equals the n-th root of b^m. Our Root Calculator is a dedicated tool for these calculations.
Any non-zero number raised to the power of 0 equals 1 (e.g. 7^0 = 1). This follows directly from the pattern of dividing consecutive powers: b^n / b^n = b^0 = 1. The single exception, 0^0, is a special case that different mathematical contexts define differently; this calculator returns 1, the most common convention.
Because 0^(-n) = 1 / 0^n = 1/0, which is division by zero — an undefined operation. The calculator flags this input instead of returning an incorrect result.