Calculate any base raised to any power, including negative and fractional exponents.
baseexponent. A negative exponent means 1/base|exponent|; a fractional exponent is a root.
A positive integer exponent is easy to picture: 2³ means 2 multiplied by itself 3 times (2 × 2 × 2 = 8). Negative and fractional exponents feel stranger at first, but they follow directly from keeping the pattern of exponent rules consistent. A negative exponent means "take the reciprocal": 2⁻² = 1/2² = 1/4 — it's not a negative number, it's a fraction, because the rule that dividing same-base powers subtracts exponents (2² ÷ 2⁴ = 2⁻²) only stays consistent if 2⁻² is defined as 1/4 rather than as -4.
A fractional exponent similarly means a root: x^(1/2) is exactly the same operation as the square root of x, and x^(1/3) is the cube root — again, this definition is what's forced by keeping the exponent rules internally consistent (since (x^(1/2))² must equal x¹ = x, exactly what squaring a square root does). Order of operations matters more with exponents than almost anywhere else in arithmetic: exponents are calculated before multiplication, division, addition or subtraction (the "E" in PEMDAS), so an expression like 2 + 3² equals 11 (calculate 3² = 9 first, then add 2), not 25 — a mistake that's easy to make when reading an expression left to right instead of by operation precedence.
A negative exponent means take the reciprocal of the positive-exponent version: 2⁻² = 1 ÷ 2² = 1/4. It's not a negative number — it's a fraction. This definition is what keeps the exponent rules (like subtracting exponents when dividing same-base powers) consistent.
A fractional exponent represents a root: x^(1/2) is the square root of x, and x^(1/3) is the cube root of x. More generally, x^(a/b) means the b-th root of x raised to the a power. This definition keeps exponent rules consistent, since squaring x^(1/2) must return x.
Because order of operations (PEMDAS/BODMAS) requires calculating exponents before addition. 3² = 9 is computed first, then 2 + 9 = 11. Reading the expression left to right and adding before exponentiating would incorrectly produce 5² = 25 — exponents always take priority over addition and subtraction.
Any nonzero number raised to the power of 0 equals 1 (x⁰ = 1). This follows from the same consistency requirement as negative exponents — dividing a power by itself (x² ÷ x² = x⁰) must equal 1, since any number divided by itself is 1.