Laws of Exponents for Real Numbers

If we have a and b as the base and m and n as the exponents, then

  1. \mathrm{a^m\times a^n=a^{m+n}}
  2. \mathrm{(a^m)^n=a^{mn}}
  3. \mathrm{\dfrac{a^m}{a^n}=a^{m-n},m>n}
  4. \mathrm{a^m\ b^m=(ab)^m}
  5. \mathrm{a^0=1}
  6. \mathrm{a^1=a}
  7. \mathrm{\dfrac{1}{a^n}=a^{-n}}

Example:

Simplify the expression  \mathrm{(2x^3y^4)\ (3xy^5)^2.}

Solution:

\mathrm{(2x^3y^4)(3xy^5)^2}

\mathrm{(2x^3y^4)(3^2x^2y^{10})}

\mathrm{18.\ x^3.x^2.y^4.y^{10}}

\mathrm{18.\ x^{3+2}.y^{4+10}}

\mathrm{18x^5y^{14}}

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