Exponent Rules: Negative Exponents and Base Operations

Negative Exponents: Basics

  • Rule: a−n=1ana^{-n} = \frac{1}{a^n}. To remove negative exponent, take reciprocal.

Separating Terms for Visualization

  • When exponent applies to multiple bases, separate: ambncpa^m b^n c^p → treat each base's exponent individually; convert negatives via reciprocal as needed.

Base Rules (Common Basis)

  • Product: aman=am+na^m a^n = a^{m+n}
  • Quotient: aman=am−n\frac{a^m}{a^n} = a^{m-n}
  • Power of a power: (ar)s=ars(a^r)^s = a^{rs}

Examples with Bases x, z, w

  • Example: x2⋅x−7/x3=x2−7−3=x−8=1x8x^2 \cdot x^{-7} / x^3 = x^{2-7-3} = x^{-8} = \frac{1}{x^8}
  • Example: If top has z2z^2 and bottom z−8z^{-8}, exponent subtraction gives z2−(−8)=z10z^{2 - (-8)} = z^{10}
  • Example: If inside has w−4w^{-4} and w4w^4, then w−4⋅w4=w0=1w^{-4} \cdot w^4 = w^{0} = 1

Distributing Exponents Across Parentheses

  • Rule: (xy)n=xnyn(xy)^n = x^n y^n
  • For a grouped expression: (apbq)r=aprbqr(a^p b^q)^r = a^{pr} b^{qr}
  • In a negative outer exponent, apply product rule to each term inside.

Working with Inside-Parentheses Exponents

  • Example: ((−4)−3)((-4)^{-3}) inside larger expression becomes 1/(−4)31/(-4)^3; more generally, (ab)c=abc(a^b)^c = a^{bc}.

Zero Exponent

  • x0=1x^0 = 1

Quick Tips

  • Leave positive exponents as-is; convert negative exponents via reciprocal.
  • Separate terms by base to visualize; combine by base using exponent rules.
  • If any base has exponent 0, that base term becomes 1.