Exercices : Puissances – 2ac

On sait que : $27=3^3$ et $9=3^2$, alors : \[\begin{array}{c}
{27^{ – x}} = {3^{ – 2x}} \times 9\\
{\left( {{3^3}} \right)^{ – x}} = {3^{ – 2x}} \times {3^2}\\
{3^{ – 3x}} = {3^{ – 2x + 2}}\end{array}\]
Et nous savons que si $a^n=a^m$ où $a>1$, alors $n=m$
Alors on a : \[\begin{array}{c}
– 3x = – 2x + 2\\
– 3x + 2x = 2\\
– x = 2\\
x = – 2\end{array}\]

\[\begin{aligned}
A &= \frac{{{a^6}}}{{{b^3}}}:\frac{{{{\left[ {{{\left( {{a^{ – 2}}{b^6}} \right)}^{ – 3}}{c^{ – 4}}} \right]}^{ – 2}}}}{{{{\left[ {{a^3}{{\left( {{b^{ – 2}}{c^{ – 4}}} \right)}^3}} \right]}^2}}}\\
A &= \frac{{{a^6}}}{{{b^3}}}:\frac{{{{\left[ {{a^6}{b^{ – 18}}{c^{ – 4}}} \right]}^{ – 2}}}}{{{{\left[ {{a^3}{b^{ – 6}}{c^{ – 12}}} \right]}^2}}}\\
A &= \frac{{{a^6}}}{{{b^3}}}:\frac{{{a^{ – 12}}{b^{36}}{c^8}}}{{{a^6}{b^{ – 12}}{c^{ – 24}}}}\\
A &= \frac{{{a^6}}}{{{b^3}}} \times \frac{{{a^6}{b^{ – 12}}{c^{ – 24}}}}{{{a^{ – 12}}{b^{36}}{c^8}}}\\
A &= {a^6}{b^{ – 3}}{a^6}{b^{ – 12}}{c^{ – 24}}{a^{12}}{b^{ – 36}}{c^{ – 8}}\\
A &= \left( {{a^6}{a^6}{a^{12}}} \right)\left( {{b^{ – 3}}{b^{ – 12}}{b^{ – 36}}} \right)\left( {{c^{ – 24}}{c^{ – 8}}} \right)\\
A &= {a^{24}}{b^{ – 51}}{c^{ – 32}}\end{aligned}\]

Corrected ici