# Exemples de potències IV

Exemples procedeixen sempre de exercicis o deures resolts que es poden convertir en exemples gràcies a les correccions.

Els que tenen numeració canviaran per uns altres de propis.

Exemples:

## 47) Passar a potència de nombres primers:

• ${\displaystyle 2^{4}\cdot 6^{2}\cdot 14^{3}=}$ ${\displaystyle 2^{4}\cdot (3\cdot 2)^{2}\cdot (2\cdot 7)^{3}=}$ ${\displaystyle 2^{4}\cdot (3^{2}\cdot 2^{2})\cdot (2^{3}\cdot 7^{3})=}$ ${\displaystyle 2^{4}\cdot 3^{2}\cdot 2^{2}\cdot 2^{3}\cdot 7^{3}=}$ ${\displaystyle 2^{4+2+3}\cdot 3^{2}\cdot 7^{3}=}$ ${\displaystyle 2^{9}\cdot 3^{2}\cdot 7^{3}}$
• ${\displaystyle 4^{3}\cdot 5^{2}\cdot 10^{-3}=}$ ${\displaystyle (2^{2})^{3}\cdot 5^{2}\cdot (2\cdot 5)^{-3}=}$ ${\displaystyle 2^{2\cdot 3}\cdot 5^{2}\cdot (2^{-3}\cdot 5^{-3})=}$ ${\displaystyle 2^{6}\cdot 5^{2}\cdot 2^{-3}\cdot 5^{-3}=}$ ${\displaystyle 2^{6-3}\cdot 5^{2-3}=}$ ${\displaystyle 2^{3}\cdot 5^{-1}}$
• ${\displaystyle {\frac {12^{3}\cdot 3^{5}}{(2^{-2})^{4}\cdot 9^{-1}}}=}$ ${\displaystyle {\frac {(2^{2}\cdot 3)^{3}\cdot 3^{5}}{(2^{-2})^{4}\cdot (3^{2})^{-1}}}=}$ ${\displaystyle {\frac {(2^{2})^{3}\cdot 3^{3}\cdot 3^{5}}{(2^{-2})^{4}\cdot (3^{2})^{-1}}}=}$ ${\displaystyle {\frac {2^{2\cdot 3}\cdot 3^{3+5}}{2^{(-2)\cdot 4}\cdot 3^{2\cdot (-1)}}}=}$ ${\displaystyle {\frac {2^{6}\cdot 3^{8}}{2^{-8}\cdot 3^{-2}}}=}$ ${\displaystyle 2^{6+8}\cdot 3^{8+2}=}$ ${\displaystyle 2^{14}\cdot 3^{10}}$
• ${\displaystyle {\frac {4^{3}\cdot 8^{-2}}{2^{5}}}=}$ ${\displaystyle {\frac {(2^{2})^{3}\cdot (2^{3})^{-2}}{2^{5}}}=}$ ${\displaystyle {\frac {2^{2\cdot 3}\cdot 2^{3\cdot (-2)}}{2^{5}}}=}$ ${\displaystyle {\frac {2^{6}\cdot 2^{-6}}{2^{5}}}=}$ ${\displaystyle 2^{6-6-5}=}$ ${\displaystyle 2^{-5}}$
• ${\displaystyle {\frac {6^{4}\cdot 4^{-3}}{8^{-2}\cdot 9^{-2}}}=}$ ${\displaystyle {\frac {(2\cdot 3)^{4}\cdot (2^{2})^{-3}}{(2^{3})^{-2}\cdot (3^{2})^{-2}}}=}$ ${\displaystyle {\frac {2^{4}\cdot 3^{4}\cdot 2^{2\cdot (-3)}}{2^{3\cdot (-2)}\cdot 3^{2\cdot (-2)}}}=}$ ${\displaystyle {\frac {2^{4}\cdot 3^{4}\cdot 2^{-6}}{2^{-6}\cdot 3^{-4}}}=}$ ${\displaystyle 2^{4-6+6}\cdot 3^{4+4}=}$ ${\displaystyle 2^{4}\cdot 3^{8}}$
• ${\displaystyle 15\cdot \left({\frac {5}{2}}\right)^{-2}=}$ ${\displaystyle 3\cdot 5\cdot {\frac {5^{-2}}{2^{-2}}}=}$ ${\displaystyle 3\cdot 5\cdot 5^{-2}\cdot 2^{2}=}$ ${\displaystyle 3\cdot 5^{1-2}\cdot 2^{2}=}$ ${\displaystyle 3\cdot 5^{-1}\cdot 2^{2}}$
• ${\displaystyle (3\cdot 5)^{-2}\cdot 3^{-4}=}$ ${\displaystyle 3^{-2}\cdot 5^{-2}\cdot 3^{-4}=}$ ${\displaystyle 3^{-2-4}\cdot 5^{-2}=}$ ${\displaystyle 3^{-6}\cdot 5^{-2}}$
• ${\displaystyle {\frac {15^{-4}\cdot 81^{-2}}{25^{-1}\cdot 45^{-2}}}=}$ ${\displaystyle {\frac {(3\cdot 5)^{-4}\cdot (3^{4})^{-2}}{(5^{2})^{-1}\cdot (3^{2}\cdot 5)^{-2}}}=}$ ${\displaystyle {\frac {3^{-4}\cdot 5^{-4}\cdot 3^{4\cdot (-2)}}{5^{2\cdot (-1)}\cdot (3^{2})^{-2}\cdot 5^{-2}}}=}$ ${\displaystyle {\frac {3^{-4}\cdot 5^{-4}\cdot 3^{-8}}{5^{-2}\cdot 3^{2\cdot (-2)}\cdot 5^{-2}}}=}$ ${\displaystyle {\frac {3^{-4}\cdot 5^{-4}\cdot 3^{-8}}{5^{-2}\cdot 3^{-4}\cdot 5^{-2}}}=}$ ${\displaystyle 3^{-4-8+4}\cdot 5^{-4+2+2}=}$ ${\displaystyle 3^{-8}\cdot 5^{0}=}$ ${\displaystyle 3^{-8}\cdot 1=}$ ${\displaystyle 3^{-8}}$

## 48) Passar a una sola potència i calcula:

• ${\displaystyle 2^{6}\cdot 5^{6}=}$ ${\displaystyle (2\cdot 5)^{6}=}$ ${\displaystyle 10^{6}=}$ ${\displaystyle 1\,000\,000}$
• ${\displaystyle {\frac {27^{2}}{9^{2}}}=}$ ${\displaystyle \left({\frac {27}{9}}\right)^{2}=}$ ${\displaystyle 3^{2}=}$ ${\displaystyle 9}$

Alternativa lenta d'aquest últim apartat:

• ${\displaystyle {\frac {27^{2}}{9^{2}}}=}$ ${\displaystyle {\frac {(3^{3})^{2}}{(3^{2})^{2}}}=}$ ${\displaystyle {\frac {3^{3\cdot 2}}{3^{2\cdot 2}}}=}$ ${\displaystyle {\frac {3^{6}}{3^{4}}}=}$ ${\displaystyle 3^{6-4}=}$ ${\displaystyle 3^{2}=}$ ${\displaystyle 9}$

## 49) Calculeu:

• ${\displaystyle {\frac {16^{3}}{32^{2}}}=}$ ${\displaystyle {\frac {(2^{4})^{3}}{(2^{5})^{2}}}=}$ ${\displaystyle {\frac {2^{4\cdot 3}}{2^{5\cdot 2}}}=}$ ${\displaystyle {\frac {2^{12}}{2^{10}}}=}$ ${\displaystyle 2^{12-10}=}$ ${\displaystyle 2^{2}=}$ ${\displaystyle 4}$