Equation field guide
Exponents: repeated multiplication and powers
An exponent tells how a base is used in multiplication. In x³, x is the base and 3 is the exponent: x³ = x × x × x.
What a power means
For a positive whole-number exponent, xⁿ is x multiplied by itself n times. A square is power 2 and a cube is power 3. The exponent applies only to the immediately preceding base unless parentheses expand it.
That distinction matters: −2² means −(2²), which is −4, while (−2)² is 4. Parentheses make a negative base unambiguous.
Zero and negative exponents
For any nonzero base, x⁰ = 1. A negative exponent gives a reciprocal: x⁻² = 1/x². These definitions preserve the multiplication rule xᵃ × xᵇ = xᵃ⁺ᵇ.
Fractional exponents connect powers and roots: x¹ᐟ² is the square root of x when working with nonnegative real values.
How to interpret it in a model
Powers often express growth, area, energy, variance, or nonlinear response. Their meaning comes from the variables and units around them.
Before calculating, identify the base and whether parentheses are present. Then apply the exponent before multiplication, division, addition, or subtraction.