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Negative Exponents and Power of a Power

Learn how negative indices and powers of powers work, like aβˆ’n=1ana^{-n} = \frac{1}{a^n} and (am)n=amn(a^m)^n = a^{mn}. Let’s get started! πŸš€

Negative Exponents and Power of a Power - introduction visual

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Laws of indices showing a⁻ᡐ = 1/aᡐ and (aᡐ)ⁿ = aᡐⁿ for negative indices and power of a powerNegative indices rules shown with examples.Power of a power rule with examples using positive and negative exponents.Power of a power rule illustrated with examples using positive, negative and fractional exponents.Laws of indices showing negative indices and power of a power, with examples including negative powers and simplification of expressions.

πŸ›ŽοΈ Negative Indices: Make a Reciprocal

  • A negative index means take the reciprocal: aβˆ’m=1ama^{-m}=\frac{1}{a^m}.
  • Example: 2βˆ’3=123=182^{-3}=\frac{1}{2^3}=\frac{1}{8}.

πŸ›ŽοΈ Negative Indices with Fractions

  • A negative index on a fraction flips it: (ab)βˆ’m=(ba)m\left(\frac{a}{b}\right)^{-m}=\left(\frac{b}{a}\right)^m.
  • Example: (23)βˆ’2=(32)2=94\left(\frac{2}{3}\right)^{-2}=\left(\frac{3}{2}\right)^2=\frac{9}{4}.

πŸ›ŽοΈ Power of a Power: Multiply the Indices

  • A power of a power means you multiply the indices: (am)n=amn(a^m)^n=a^{mn}.
  • Example: (23)5=23Γ—5=215(2^3)^5=2^{3\times5}=2^{15}.

πŸ›ŽοΈ Power of a Power with Negative Indices

  • You still multiply the indices, so two negatives make a positive index.
  • Example: (3βˆ’2)βˆ’2=3(βˆ’2)Γ—(βˆ’2)=34=81(3^{-2})^{-2}=3^{(-2)\times(-2)}=3^4=81.

πŸ›ŽοΈ Laws of Indices Summary

  • If the base is a fraction and the index is negative, you flip the fraction to make the index positive.
  • A power of a power means **multiply the indices, even if an index is negative.

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