I can follow the rules mechanically but I don't understand what x^(−2) or x^(1/2) *is*.
Accepted answer
Both fall out of keeping one pattern consistent: **each step down in the exponent divides by the base.**
`2³ = 8`, `2² = 4`, `2¹ = 2`, `2⁰ = 1` — halving each time. Continue and `2⁻¹ = 1/2`, `2⁻² = 1/4`.
So **a negative exponent means reciprocal**: `x⁻ⁿ = 1/xⁿ`. It does *not* make the answer negative — `2⁻² = 1/4`, not `−4`.
**Fractional exponents are roots.** Since `x^(1/2) · x^(1/2) = x^1 = x`, the thing that multiplies by itself to give `x` is the square root: `x^(1/2) = √x`. Generally `x^(m/n) = ⁿ√(xᵐ)` — denominator is the root, numerator is the power.
Combining: `8^(−2/3) = 1/8^(2/3) = 1/(∛8)² = 1/4`. Take the root first to keep the numbers small.