Calculs avec des puissances
A = \(\dfrac{49\times(-2)^{5}\times(-3)^{-2}}{-7^{3}\times 16\times 3^{-3}}\)
Corrigé
\[\begin{aligned}
A&=\dfrac{49\times(-2)^{5}\times(-3)^{-2}}{-7^{3}\times 16\times 3^{-3}}\\
A&=\dfrac{7^{2}\times 2^{5}\times 3^{-2}}{7^{3}\times 2^{4}\times 3^{-3}}\\
A&=7^{2-3}\times 2^{5-4}\times 3^{-2-(-3)}\\
A&=7^{-1}\times 2\times 3\\
A&=\dfrac{6}{7}
\end{aligned}\]
B = \(\dfrac{(-5)^{4}\times 7^{2}\times(-2)^{-3}}{(-4)^{4}\times(-1)^{5}\times 25}\)
Corrigé
\[\begin{aligned}
B&=\dfrac{(-5)^{4}\times 7^{2}\times(-2)^{-3}}{(-4)^{4}\times(-1)^{5}\times 25}\\
B&=\dfrac{5^{4}\times 7^{2}\times 2^{-3}}{2^{8}\times 5^{2}}\\
B&=5^{4-2}\times 7^{2}\times 2^{-3-8}\\
B&=5^{2}\times 7^{2}\times 2^{-11}\\
B&=\dfrac{1225}{2048}
\end{aligned}\]
C = \(\left(\dfrac{(a^{2}b^{4})^{2}}{a^{3}}\right)^{-3}\)
Corrigé
\[\begin{aligned}
C&=\left(\dfrac{(a^{2}b^{4})^{2}}{a^{3}}\right)^{-3}\\
C&=\left(\dfrac{a^{4}b^{8}}{a^{3}}\right)^{-3}\\
C&=\left(a b^{8}\right)^{-3}\\
C&=a^{-3}b^{-24}\\
C&=\dfrac{1}{a^{3}b^{24}}
\end{aligned}\]
D = \(0,0000000005\times 1004000000\)
Corrigé
\[\begin{aligned}
D&=0,0000000005\times 1004000000\\
D&=\left(5\times 10^{-10}\right)\times\left(1,004\times 10^{9}\right)\\
D&=5,02\times 10^{-1}\\
D&=0,502
\end{aligned}\]
E = \(\dfrac{2^{3}}{3^{4}}\div\dfrac{2^{2}}{3^{5}}\)
Corrigé
\[\begin{aligned}
E&=\dfrac{2^{3}}{3^{4}}\div\dfrac{2^{2}}{3^{5}}\\
E&=\dfrac{2^{3}}{3^{4}}\times\dfrac{3^{5}}{2^{2}}\\
E&=2^{3-2}\times 3^{5-4}\\
E&=2\times 3\\
E&=6
\end{aligned}\]
F = \(\dfrac{(a^{2}b)^{3}}{(-a)(-b)^{2}}\)
Corrigé
\[\begin{aligned}
F&=\dfrac{(a^{2}b)^{3}}{(-a)(-b)^{2}}\\
F&=\dfrac{a^{6}b^{3}}{-ab^{2}}\\
F&=-a^{6-1}b^{3-2}\\
F&=-a^{5}b
\end{aligned}\]
G = \(\left(\dfrac{4^{-2}\times 8^{4}}{90^{7}\times 30^{-2}}\right)^{3}\)
Corrigé
\[\begin{aligned}
G&=\left(\dfrac{4^{-2}\times 8^{4}}{90^{7}\times 30^{-2}}\right)^{3}\\
G&=\left(\dfrac{2^{-4}\times 2^{12}}{2^{7}\times 3^{14}\times 5^{7}\times 2^{-2}\times 3^{-2}\times 5^{-2}}\right)^{3}\\
G&=\left(\dfrac{2^{8}}{2^{5}\times 3^{12}\times 5^{5}}\right)^{3}\\
G&=\left(\dfrac{2^{3}}{3^{12}\times 5^{5}}\right)^{3}\\
G&=\dfrac{2^{9}}{3^{36}\times 5^{15}}
\end{aligned}\]
H = \(\left(\dfrac{5^{5}\times 24^{-3}}{(100^{-7}\times 15^{6})^{4}}\right)^{2}\)
Corrigé
\[\begin{aligned}
H&=\left(\dfrac{5^{5}\times 24^{-3}}{(100^{-7}\times 15^{6})^{4}}\right)^{2}\\
H&=\left(\dfrac{5^{5}\times 2^{-9}\times 3^{-3}}{(2^{-14}\times 5^{-14}\times 3^{6}\times 5^{6})^{4}}\right)^{2}\\
H&=\left(\dfrac{5^{5}\times 2^{-9}\times 3^{-3}}{(2^{-14}\times 3^{6}\times 5^{-8})^{4}}\right)^{2}\\
H&=\left(\dfrac{5^{5}\times 2^{-9}\times 3^{-3}}{2^{-56}\times 3^{24}\times 5^{-32}}\right)^{2}\\
H&=\left(2^{47}\times 3^{-27}\times 5^{37}\right)^{2}\\
H&=2^{94}\times 3^{-54}\times 5^{74}\\
H&=\dfrac{2^{94}\times 5^{74}}{3^{54}}
\end{aligned}\]
I = \(\dfrac{2^{2}\times 10^{-10}\times 2^{7}\times 10^{-6}}{32\times 10^{-15}}\)
Corrigé
\[\begin{aligned}
I&=\dfrac{2^{2}\times 10^{-10}\times 2^{7}\times 10^{-6}}{32\times 10^{-15}}\\
I&=\dfrac{2^{9}\times 10^{-16}}{2^{5}\times 10^{-15}}\\
I&=2^{9-5}\times 10^{-16-(-15)}\\
I&=2^{4}\times 10^{-1}\\
I&=\dfrac{16}{10}\\
I&=\dfrac{8}{5}
\end{aligned}\]
J = \(\left(\dfrac{a^{3}b^{-2}}{a^{4}b^{-3}}\right)^{-2}\times\dfrac{(3a^{2}b^{3})^{3}}{(2^{-1}ab)^{2}}\)
Corrigé
\[\begin{aligned}
J&=\left(\dfrac{a^{3}b^{-2}}{a^{4}b^{-3}}\right)^{-2}\times\dfrac{(3a^{2}b^{3})^{3}}{(2^{-1}ab)^{2}}\\
J&=\left(a^{-1}b\right)^{-2}\times\dfrac{27a^{6}b^{9}}{2^{-2}a^{2}b^{2}}\\
J&=a^{2}b^{-2}\times 27\times 2^{2}\times a^{4}b^{7}\\
J&=108\,a^{6}b^{5}
\end{aligned}\]
K = \(\dfrac{5^{3}\times 3^{8}\times 5^{2}}{125\times 5^{2}\times 81\times 7^{0}}\)
Corrigé
\[\begin{aligned}
K&=\dfrac{5^{3}\times 3^{8}\times 5^{2}}{125\times 5^{2}\times 81\times 7^{0}}\\
K&=\dfrac{5^{5}\times 3^{8}}{5^{5}\times 3^{4}}\\
K&=3^{8-4}\\
K&=3^{4}\\
K&=81
\end{aligned}\]
L = \(\dfrac{0,9\times 7\times 10^{-1}\times 250}{14\times 10^{3}\times 0,5\times 10^{-2}}\)
Corrigé
\[\begin{aligned}
L&=\dfrac{0,9\times 7\times 10^{-1}\times 250}{14\times 10^{3}\times 0,5\times 10^{-2}}\\
L&=\dfrac{9\times 10^{-1}\times 7\times 10^{-1}\times 2\times 5^{3}}{2\times 7\times 10^{3}\times 5\times 10^{-1}\times 10^{-2}}\\
L&=\dfrac{9\times 7\times 2\times 5^{3}\times 10^{-2}}{2\times 7\times 5\times 10^{0}}\\
L&=9\times 5^{2}\times 10^{-2}\\
L&=\dfrac{9\times 25}{100}\\
L&=\dfrac{9}{4}
\end{aligned}\]
M = \(\dfrac{(56^{8}\times 81^{-2}\times 25^{7})^{3}}{(50^{5}\times 700^{3})^{4}}\)
Corrigé
\[\begin{aligned}
M&=\dfrac{(56^{8}\times 81^{-2}\times 25^{7})^{3}}{(50^{5}\times 700^{3})^{4}}\\
M&=\dfrac{(2^{24}\times 7^{8}\times 3^{-8}\times 5^{14})^{3}}{(2^{5}\times 5^{10}\times 2^{6}\times 5^{6}\times 7^{3})^{4}}\\
M&=\dfrac{2^{72}\times 7^{24}\times 3^{-24}\times 5^{42}}{(2^{11}\times 5^{16}\times 7^{3})^{4}}\\
M&=\dfrac{2^{72}\times 7^{24}\times 3^{-24}\times 5^{42}}{2^{44}\times 5^{64}\times 7^{12}}\\
M&=2^{72-44}\times 7^{24-12}\times 3^{-24}\times 5^{42-64}\\
M&=2^{28}\times 7^{12}\times 3^{-24}\times 5^{-22}\\
M&=\dfrac{2^{28}\times 7^{12}}{3^{24}\times 5^{22}}
\end{aligned}\]
N = \(\dfrac{0,04\times 2^{-2}\times(10^{-2})^{3}\times 10^{2}}{3\times 10^{-8}\times 10^{-2}}\)
Corrigé
\[\begin{aligned}
N&=\dfrac{0,04\times 2^{-2}\times(10^{-2})^{3}\times 10^{2}}{3\times 10^{-8}\times 10^{-2}}\\
N&=\dfrac{2^{2}\times 10^{-2}\times 2^{-2}\times 10^{-6}\times 10^{2}}{3\times 10^{-10}}\\
N&=\dfrac{10^{-6}}{3\times 10^{-10}}\\
N&=\dfrac{1}{3}\times 10^{-6-(-10)}\\
N&=\dfrac{1}{3}\times 10^{4}\\
N&=\dfrac{10000}{3}
\end{aligned}\]
O = \(\dfrac{25\times(10^{2})^{-5}\times 121}{11\times 75\times 10^{-9}}\)
Corrigé
\[\begin{aligned}
O&=\dfrac{25\times(10^{2})^{-5}\times 121}{11\times 75\times 10^{-9}}\\
O&=\dfrac{5^{2}\times 11^{2}\times 10^{-10}}{11\times 3\times 5^{2}\times 10^{-9}}\\
O&=11^{2-1}\times 5^{2-2}\times 10^{-10-(-9)}\div 3\\
O&=11\times 10^{-1}\div 3\\
O&=\dfrac{11}{30}
\end{aligned}\]
P = \(\dfrac{9^{n+1}+9^{n}}{3^{2n+1}-3^{2n}}\) \((n \in \mathbb{N})\)
Corrigé
\[\begin{aligned}
P&=\dfrac{9^{n+1}+9^{n}}{3^{2n+1}-3^{2n}}\\
P&=\dfrac{3^{2n+2}+3^{2n}}{3^{2n+1}-3^{2n}}\\
P&=\dfrac{3^{2n}\left(3^{2}+1\right)}{3^{2n}\left(3-1\right)}\\
P&=\dfrac{10}{2}\\
P&=5
\end{aligned}\]
Q = \(\dfrac{(ab^{2})^{2}(ab^{-1})^{3}(a^{2}b)^{-2}}{a^{2}c^{-5}(a^{-1}bc^{2})^{3}}\)
Corrigé
\[\begin{aligned}
Q&=\dfrac{(ab^{2})^{2}(ab^{-1})^{3}(a^{2}b)^{-2}}{a^{2}c^{-5}(a^{-1}bc^{2})^{3}}\\
Q&=\dfrac{a^{2}b^{4}\times a^{3}b^{-3}\times a^{-4}b^{-2}}{a^{2}c^{-5}\times a^{-3}b^{3}c^{6}}\\
Q&=\dfrac{a^{2+3-4}b^{4-3-2}}{a^{2-3}b^{3}c^{-5+6}}\\
Q&=\dfrac{a\,b^{-1}}{a^{-1}b^{3}c}\\
Q&=a^{1-(-1)}b^{-1-3}c^{-1}\\
Q&=a^{2}b^{-4}c^{-1}\\
Q&=\dfrac{a^{2}}{b^{4}c}
\end{aligned}\]
R = \(\dfrac{(ab^{-2}c^{3})^{4}(a^{4}b^{5}c^{-6})^{-2}}{(a^{-7}b^{8}c^{7})^{3}(a^{6}b^{5}c^{4})^{2}}\)
Corrigé
\[\begin{aligned}
R&=\dfrac{(ab^{-2}c^{3})^{4}(a^{4}b^{5}c^{-6})^{-2}}{(a^{-7}b^{8}c^{7})^{3}(a^{6}b^{5}c^{4})^{2}}\\
R&=\dfrac{a^{4}b^{-8}c^{12}\times a^{-8}b^{-10}c^{12}}{a^{-21}b^{24}c^{21}\times a^{12}b^{10}c^{8}}\\
R&=\dfrac{a^{4-8}b^{-8-10}c^{12+12}}{a^{-21+12}b^{24+10}c^{21+8}}\\
R&=\dfrac{a^{-4}b^{-18}c^{24}}{a^{-9}b^{34}c^{29}}\\
R&=a^{-4-(-9)}b^{-18-34}c^{24-29}\\
R&=a^{5}b^{-52}c^{-5}\\
R&=\dfrac{a^{5}}{b^{52}c^{5}}
\end{aligned}\]