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......@@ -232,46 +232,46 @@ Réduire et ordonner les expressions algébriques ci-dessous.
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## :red_circle: Niveau C - recherché en fin de programme
Réduire et ordonner les expressions algébriques ci-dessous.
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1. Réduire et ordonner : $\dfrac{2a^2b}{3} \cdot \dfrac{9ab^3}{4a^2} \cdot \dfrac{6a^3}{5b^2} =$
1. $\quad \dfrac{2a^2b}{3} \cdot \dfrac{9ab^3}{4a^2} \cdot \dfrac{6a^3}{5b^2} =$
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2. Réduire et ordonner : $(3x^2y^3)^2 \cdot \dfrac{2xy}{9x^3y^4} \cdot (-4x^2y) =$
2. $\quad (3x^2y^3)^2 \cdot \dfrac{2xy}{9x^3y^4} \cdot (-4x^2y) =$
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3. Réduire et ordonner : $\left(\dfrac{2ab^2}{3}\right)^3 \cdot \dfrac{9a^2}{4b^3} \cdot \dfrac{2b}{a^2} =$
3. $\quad \left(\dfrac{2ab^2}{3}\right)^3 \cdot \dfrac{9a^2}{4b^3} \cdot \dfrac{2b}{a^2} =$
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4. Réduire et ordonner : $\dfrac{4x^3y^2z}{3xy} \cdot \dfrac{6xyz^2}{8x^2z} \cdot \dfrac{9x^2y}{2yz^2} =$
4. $\quad \dfrac{4x^3y^2z}{3xy} \cdot \dfrac{6xyz^2}{8x^2z} \cdot \dfrac{9x^2y}{2yz^2} =$
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5. Réduire et ordonner : $2x^3y - 3x^2y^2 + xy^3 - x^3y + 2x^2y^2 - 3xy^3 + 5x^3y - xy^3 =$
5. $\quad 2x^3y - 3x^2y^2 + xy^3 - x^3y + 2x^2y^2 - 3xy^3 + 5x^3y - xy^3 =$
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6. Réduire et ordonner : $\dfrac{3m^2n}{2} + \dfrac{4mn^2}{3} - \dfrac{2m^3}{5} + \dfrac{5m^2n}{4} - \dfrac{8mn^2}{6} + \dfrac{m^3}{10} =$
6. $\quad \dfrac{3m^2n}{2} + \dfrac{4mn^2}{3} - \dfrac{2m^3}{5} + \dfrac{5m^2n}{4} - \dfrac{8mn^2}{6} + \dfrac{m^3}{10} =$
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7. Réduire et ordonner : $a^2b(2ab^2) - 3ab(a^2b) + 5a^3b^2 - 2a^2b \cdot ab + 4ab \cdot a^2b =$
7. $\quad a^2b(2ab^2) - 3ab(a^2b) + 5a^3b^2 - 2a^2b \cdot ab + 4ab \cdot a^2b =$
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<details>
<summary>:check_mark_button: Solutions Niveau C</summary>
1. $\dfrac{2a^2b}{3} \cdot \dfrac{9ab^3}{4a^2} \cdot \dfrac{6a^3}{5b^2} = \dfrac{2 \cdot 9 \cdot 6 \cdot a^{2+1+3} \cdot b^{1+3-2}}{3 \cdot 4 \cdot 5 \cdot a^2} = \dfrac{108a^6b^2}{60a^2} = \dfrac{108a^4b^2}{60} = \dfrac{9a^4b^2}{5}$
1. $\quad \dfrac{2a^2b}{3} \cdot \dfrac{9ab^3}{4a^2} \cdot \dfrac{6a^3}{5b^2} = \dfrac{2 \cdot 9 \cdot 6 \cdot a^{2+1+3} \cdot b^{1+3-2}}{3 \cdot 4 \cdot 5 \cdot a^2} = \dfrac{108a^6b^2}{60a^2} = \dfrac{108a^4b^2}{60} = \boxed{\dfrac{9a^4b^2}{5}}$
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2. $(3x^2y^3)^2 \cdot \dfrac{2xy}{9x^3y^4} \cdot (-4x^2y) = 9x^4y^6 \cdot \dfrac{2xy}{9x^3y^4} \cdot (-4x^2y) = 9x^4y^6 \cdot \dfrac{-8x^3y^2}{9x^3y^4} = 9x^4y^6 \cdot \dfrac{-8}{9y^2} = -8x^4y^4$
2. $\quad (3x^2y^3)^2 \cdot \dfrac{2xy}{9x^3y^4} \cdot (-4x^2y) = 9x^4y^6 \cdot \dfrac{2xy}{9x^3y^4} \cdot (-4x^2y) = 9x^4y^6 \cdot \dfrac{-8x^3y^2}{9x^3y^4} = 9x^4y^6 \cdot \dfrac{-8}{9y^2} = \boxed{-8x^4y^4}$
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3. $\left(\dfrac{2ab^2}{3}\right)^3 \cdot \dfrac{9a^2}{4b^3} \cdot \dfrac{2b}{a^2} = \dfrac{8a^3b^6}{27} \cdot \dfrac{9a^2}{4b^3} \cdot \dfrac{2b}{a^2} = \dfrac{8 \cdot 9 \cdot 2 \cdot a^{3+2-2} \cdot b^{6-3+1}}{27 \cdot 4} = \dfrac{144a^3b^4}{108} = \dfrac{4a^3b^4}{3}$
3. $\quad \left(\dfrac{2ab^2}{3}\right)^3 \cdot \dfrac{9a^2}{4b^3} \cdot \dfrac{2b}{a^2} = \dfrac{8a^3b^6}{27} \cdot \dfrac{9a^2}{4b^3} \cdot \dfrac{2b}{a^2} = \dfrac{8 \cdot 9 \cdot 2 \cdot a^{3+2-2} \cdot b^{6-3+1}}{27 \cdot 4} = \dfrac{144a^3b^4}{108} = \boxed{\dfrac{4a^3b^4}{3}}$
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4. $\dfrac{4x^3y^2z}{3xy} \cdot \dfrac{6xyz^2}{8x^2z} \cdot \dfrac{9x^2y}{2yz^2} = \dfrac{4 \cdot 6 \cdot 9 \cdot x^{3+1+2} \cdot y^{2+1+1} \cdot z^{1+2-2}}{3 \cdot 8 \cdot 2 \cdot x^{1+2} \cdot y^{1+1} \cdot z^{1+2}} = \dfrac{216x^6y^4z}{48x^3y^2z^3} = \dfrac{216x^3y^2}{48z^2} = \dfrac{9x^3y^2}{2z^2}$
4. $\quad \dfrac{4x^3y^2z}{3xy} \cdot \dfrac{6xyz^2}{8x^2z} \cdot \dfrac{9x^2y}{2yz^2} = \dfrac{4 \cdot 6 \cdot 9 \cdot x^{3+1+2} \cdot y^{2+1+1} \cdot z^{1+2-2}}{3 \cdot 8 \cdot 2 \cdot x^{1+2} \cdot y^{1+1} \cdot z^{1+2}} = \dfrac{216x^6y^4z}{48x^3y^2z^3} = \dfrac{216x^3y^2}{48z^2} = \boxed{\dfrac{9x^3y^2}{2z^2}}$
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5. $2x^3y - 3x^2y^2 + xy^3 - x^3y + 2x^2y^2 - 3xy^3 + 5x^3y - xy^3$
$= (2-1+5)x^3y + (-3+2)x^2y^2 + (1-3-1)xy^3 = 6x^3y - x^2y^2 - 3xy^3$
5. $\quad 2x^3y - 3x^2y^2 + xy^3 - x^3y + 2x^2y^2 - 3xy^3 + 5x^3y - xy^3$
$= (2-1+5)x^3y + (-3+2)x^2y^2 + (1-3-1)xy^3 = \boxe{6x^3y - x^2y^2 - 3xy^3}$
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6. $\dfrac{3m^2n}{2} + \dfrac{4mn^2}{3} - \dfrac{2m^3}{5} + \dfrac{5m^2n}{4} - \dfrac{8mn^2}{6} + \dfrac{m^3}{10}$
6. $\quad \dfrac{3m^2n}{2} + \dfrac{4mn^2}{3} - \dfrac{2m^3}{5} + \dfrac{5m^2n}{4} - \dfrac{8mn^2}{6} + \dfrac{m^3}{10}$
$= m^2n\left(\dfrac{3}{2} + \dfrac{5}{4}\right) + mn^2\left(\dfrac{4}{3} - \dfrac{8}{6}\right) + m^3\left(-\dfrac{2}{5} + \dfrac{1}{10}\right)$
$= m^2n\left(\dfrac{6+5}{4}\right) + mn^2\left(\dfrac{4-4}{3}\right) + m^3\left(\dfrac{-4+1}{10}\right)$
$= \dfrac{11m^2n}{4} + 0 - \dfrac{3m^3}{10} = -\dfrac{3m^3}{10} + \dfrac{11m^2n}{4}$
$= \dfrac{11m^2n}{4} + 0 - \dfrac{3m^3}{10} = \boxed{-\dfrac{3m^3}{10} + \dfrac{11m^2n}{4}}$
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7. $a^2b(2ab^2) - 3ab(a^2b) + 5a^3b^2 - 2a^2b \cdot ab + 4ab \cdot a^2b$
7. $\quad a^2b(2ab^2) - 3ab(a^2b) + 5a^3b^2 - 2a^2b \cdot ab + 4ab \cdot a^2b$
$= 2a^3b^3 - 3a^3b^2 + 5a^3b^2 - 2a^3b^2 + 4a^3b^2$
$= 2a^3b^3 + (-3+5-2+4)a^3b^2 = 2a^3b^3 + 4a^3b^2 = 2a^3b^2(b + 2)$
$= 2a^3b^3 + (-3+5-2+4)a^3b^2 = \boxed{2a^3b^3 + 4a^3b^2}$
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