Ensembles de nombres ℕ, ℤ, ℚ, 𝔻, ℝ
5. 📚 Exercices – Calcul littéral et puissances
Tronc Commun Sciences BIOF
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📌 Exercice 12
Soient a et b deux entiers naturels, on pose \( Z = a^2 + 2ab + b^2 + a + b + 1 \).
Écrire Z sous la forme d'une somme de carrés de trois entiers naturels.
📌 Exercice 13
Soient \( x, y \) et \( z \) des réels non nuls. Simplifier les expressions suivantes :
\[ X = \frac{(x^2y)^{-3}xz^2}{xy^{-3}}, \quad Y = \frac{(-x)^2x^2y}{2y^{-1}}, \quad Z = \frac{(x^2y)^{-3}z^5x^4}{(xy^2)^2y^{-1}}, \quad T = \frac{(-x)^5yz^{-2}}{y^3(-z)^{-3}x^2} \]
📌 Exercice 14
Développer les expressions suivantes :
\[ A = (x^2 + x + 1)(2x - 5), \quad B = (7x - 3y)^2, \quad C = (x - 1)(x + 2)(x - 2), \quad D = (5a + 3)^3 \] \[ E = (2x - 5y)^3, \quad F = (3 + x + a)^2, \quad G = (2\sqrt{3}p + \sqrt{5q})^3, \quad H = \left(t - \frac{1}{t}\right)^3 \]
📌 Exercice 15
Factoriser les expressions suivantes :
\[ A = a^2 - 2ab + b^2 - c^2, \quad B = (3x^2 - 3) + (x^2 - 2x + 1), \quad C = x^{16} - 16, \quad D = 2\sqrt{2}x^3 + 27 \] \[ E = x^{12} - 2x^6 + 1, \quad F = x^5 + x^3 - x^2 - 1, \quad I = 125a^3 + 64, \quad J = 2(x + 7)(x + 5) - (x^2 - 25) \] \[ K = 6x(x - 2) - 3x^2 + 12, \quad L = x^3 - 8 - 5(x - 2), \quad M = 64 - (5x - 7)^3, \quad N = (x - 7y)^3 + 27y^3 \] \[ U = x^3 - 8 + 4(x^2 - 4) - 3x + 6, \quad V = x^3 + 1 + 2(x^2 - 1) - (x + 1) \]
📌 Exercice 16
a est un nombre réel non nul, on pose : \( A = a + \frac{1}{a} \).
Calculer en fonction de \( A \) les nombres suivants :
\[ a^2 + \frac{1}{a^2}, \quad a^3 + \frac{1}{a^3}, \quad a^4 + \frac{1}{a^4} \]
📌 Exercice 17
Déterminer trois nombres a, b et c tels que :
\[ 2^a \times 3^b \times 5^c = 648000 \]
📌 Exercice 18
Soient a et b deux nombres réels tels que :
\[ 2(a^2 + b^2) = 5ab \]
Calculer A sachant que : \( A = \frac{a - b}{a + b} \).
📌 Exercice 19
a, b et c des nombres réels tels que : \( abc = 1 \).
Montrer que :
\[ \frac{a}{ab + a + 1} + \frac{b}{bc + b + 1} + \frac{c}{ca + c + 1} = 1 \]
📌 Exercice 20
Donner l'écriture scientifique de chacun des nombres suivants :
\[ A = 0,0004651, \quad B = 7560000000, \quad C = 450087 + 23 \times 10^4, \quad D = 0,0018 + 7 \times 10^{-4} \] \[ E = 17,001 \times 10^8, \quad c = 299792458, \quad e = 1602,1892 \times 10^{-22}, \quad g = 980,665 \times 10^{-2} \] \[ u = 166,0565 \times 10^{-29}, \quad N_A = 60220,45 \times 10^{19}, \quad h = 0,6626176 \times 10^{-33} \]
📌 Exercice 21
Soient a et b deux nombres réels non nuls. Calculer
\[ \frac{a^2b(a^2b-1)^4a^{-3}b^2}{ab^2(a^2b-1)^2(a^2b^3)(a^2b^3)^3} \]
pour \( a = 10^{-3} \) et \( b = -10^{-2} \).
📌 Exercice 22
a, b et c des nombres réels non nuls tels que : \( ab + bc + ca = 0 \).
Calculer
\[ \frac{b+c}{a} + \frac{c+a}{b} + \frac{a+b}{c} \]
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✅ Exercice 12
Énoncé : \( Z = a^2 + 2ab + b^2 + a + b + 1 \)
Écrire Z sous la forme d'une somme de carrés de trois entiers naturels.
\[ Z = (a^2 + 2ab + b^2) + (a + b) + 1 \] \[ Z = (a + b)^2 + (a + b) + 1 \] \[ Z = (a + b)^2 + (a + b) + \frac{1}{4} + \frac{3}{4} \] \[ Z = \left(a + b + \frac{1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2 \]
Si on pose \( a + b = k \), alors \( Z = k^2 + k + 1 \).
On peut aussi écrire : \( Z = (a+b)^2 + (a+b) + 1 \).
✅ Exercice 13
Énoncé : Simplifier les expressions.
\( X = \frac{(x^2y)^{-3}xz^2}{xy^{-3}} \)
\[ X = \frac{x^{-6}y^{-3} \cdot x \cdot z^2}{x \cdot y^{-3}} = \frac{x^{-5}y^{-3}z^2}{x y^{-3}} = x^{-6}z^2 = \frac{z^2}{x^6} \]
\( Y = \frac{(-x)^2x^2y}{2y^{-1}} \)
\[ Y = \frac{x^2 \cdot x^2 \cdot y}{2 \cdot y^{-1}} = \frac{x^4 y \cdot y}{2} = \frac{x^4 y^2}{2} \]
\( Z = \frac{(x^2y)^{-3}z^5x^4}{(xy^2)^2y^{-1}} \)
\[ Z = \frac{x^{-6}y^{-3}z^5x^4}{x^2y^4y^{-1}} = \frac{x^{-2}y^{-3}z^5}{x^2y^3} = x^{-4}y^{-6}z^5 = \frac{z^5}{x^4 y^6} \]
\( T = \frac{(-x)^5yz^{-2}}{y^3(-z)^{-3}x^2} \)
\[ T = \frac{-x^5yz^{-2}}{y^3(-z)^{-3}x^2} = \frac{-x^5yz^{-2} \cdot (-z)^3}{y^3x^2} = \frac{-x^5yz^{-2} \cdot (-z^3)}{y^3x^2} \] \[ T = \frac{x^5 y z^{-2} z^3}{y^3 x^2} = \frac{x^3 y z}{y^3} = \frac{x^3 z}{y^2} \]
✅ Exercice 14
Énoncé : Développer les expressions.
\( A = (x^2 + x + 1)(2x - 5) \)
\[ A = 2x^3 - 5x^2 + 2x^2 - 5x + 2x - 5 = 2x^3 - 3x^2 - 3x - 5 \]
\( B = (7x - 3y)^2 \)
\[ B = 49x^2 - 42xy + 9y^2 \]
\( C = (x - 1)(x + 2)(x - 2) \)
\[ C = (x - 1)(x^2 - 4) = x^3 - 4x - x^2 + 4 = x^3 - x^2 - 4x + 4 \]
\( D = (5a + 3)^3 \)
\[ D = 125a^3 + 3 \cdot 25a^2 \cdot 3 + 3 \cdot 5a \cdot 9 + 27 = 125a^3 + 225a^2 + 135a + 27 \]
\( E = (2x - 5y)^3 \)
\[ E = 8x^3 - 3 \cdot 4x^2 \cdot 5y + 3 \cdot 2x \cdot 25y^2 - 125y^3 = 8x^3 - 60x^2y + 150xy^2 - 125y^3 \]
\( F = (3 + x + a)^2 \)
\[ F = (3 + x + a)^2 = 9 + x^2 + a^2 + 6x + 6a + 2ax \]
\( G = (2\sqrt{3}p + \sqrt{5q})^3 \)
\[ G = 8 \cdot 3\sqrt{3}p^3 + 3 \cdot (2\sqrt{3}p)^2 \cdot \sqrt{5q} + 3 \cdot (2\sqrt{3}p)(\sqrt{5q})^2 + (\sqrt{5q})^3 \] \[ G = 24\sqrt{3}p^3 + 3 \cdot 12p^2 \cdot \sqrt{5q} + 6\sqrt{3}p \cdot 5q + 5q\sqrt{5q} \] \[ G = 24\sqrt{3}p^3 + 36p^2\sqrt{5q} + 30\sqrt{3}pq + 5q\sqrt{5q} \]
\( H = \left(t - \frac{1}{t}\right)^3 \)
\[ H = t^3 - 3t^2 \cdot \frac{1}{t} + 3t \cdot \frac{1}{t^2} - \frac{1}{t^3} = t^3 - 3t + \frac{3}{t} - \frac{1}{t^3} \]
✅ Exercice 15
Énoncé : Factoriser les expressions.
\( A = a^2 - 2ab + b^2 - c^2 \)
\[ A = (a - b)^2 - c^2 = (a - b - c)(a - b + c) \]
\( B = (3x^2 - 3) + (x^2 - 2x + 1) \)
\[ B = 3(x^2 - 1) + (x - 1)^2 = 3(x - 1)(x + 1) + (x - 1)^2 = (x - 1)[3(x + 1) + (x - 1)] \] \[ B = (x - 1)(3x + 3 + x - 1) = (x - 1)(4x + 2) = 2(x - 1)(2x + 1) \]
\( C = x^{16} - 16 \)
\[ C = (x^8 - 4)(x^8 + 4) = (x^4 - 2)(x^4 + 2)(x^8 + 4) \]
\( D = 2\sqrt{2}x^3 + 27 \)
\[ D = (\sqrt{2}x)^3 + 3^3 = (\sqrt{2}x + 3)(2x^2 - 3\sqrt{2}x + 9) \]
\( E = x^{12} - 2x^6 + 1 \)
\[ E = (x^6 - 1)^2 = (x^3 - 1)^2(x^3 + 1)^2 = (x - 1)^2(x^2 + x + 1)^2(x + 1)^2(x^2 - x + 1)^2 \]
\( F = x^5 + x^3 - x^2 - 1 \)
\[ F = x^3(x^2 + 1) - (x^2 + 1) = (x^2 + 1)(x^3 - 1) = (x^2 + 1)(x - 1)(x^2 + x + 1) \]
\( I = 125a^3 + 64 \)
\[ I = (5a)^3 + 4^3 = (5a + 4)(25a^2 - 20a + 16) \]
\( J = 2(x + 7)(x + 5) - (x^2 - 25) \)
\[ J = 2(x + 7)(x + 5) - (x - 5)(x + 5) = (x + 5)[2(x + 7) - (x - 5)] \] \[ J = (x + 5)(2x + 14 - x + 5) = (x + 5)(x + 19) \]
\( K = 6x(x - 2) - 3x^2 + 12 \)
\[ K = 6x(x - 2) - 3(x^2 - 4) = 6x(x - 2) - 3(x - 2)(x + 2) \] \[ K = (x - 2)(6x - 3x - 6) = (x - 2)(3x - 6) = 3(x - 2)(x - 2) = 3(x - 2)^2 \]
\( L = x^3 - 8 - 5(x - 2) \)
\[ L = (x - 2)(x^2 + 2x + 4) - 5(x - 2) = (x - 2)(x^2 + 2x + 4 - 5) \] \[ L = (x - 2)(x^2 + 2x - 1) \]
\( M = 64 - (5x - 7)^3 \)
\[ M = 4^3 - (5x - 7)^3 = (4 - (5x - 7))(16 + 4(5x - 7) + (5x - 7)^2) \] \[ M = (-5x + 11)(16 + 20x - 28 + 25x^2 - 70x + 49) \] \[ M = (-5x + 11)(25x^2 - 50x + 37) \]
\( N = (x - 7y)^3 + 27y^3 \)
\[ N = (x - 7y)^3 + (3y)^3 = (x - 7y + 3y)((x - 7y)^2 - 3y(x - 7y) + 9y^2) \] \[ N = (x - 4y)(x^2 - 14xy + 49y^2 - 3xy + 21y^2 + 9y^2) \] \[ N = (x - 4y)(x^2 - 17xy + 79y^2) \]
\( U = x^3 - 8 + 4(x^2 - 4) - 3x + 6 \)
\[ U = (x - 2)(x^2 + 2x + 4) + 4(x - 2)(x + 2) - 3(x - 2) \] \[ U = (x - 2)[x^2 + 2x + 4 + 4x + 8 - 3] = (x - 2)(x^2 + 6x + 9) \] \[ U = (x - 2)(x + 3)^2 \]
\( V = x^3 + 1 + 2(x^2 - 1) - (x + 1) \)
\[ V = (x + 1)(x^2 - x + 1) + 2(x - 1)(x + 1) - (x + 1) \] \[ V = (x + 1)[x^2 - x + 1 + 2x - 2 - 1] = (x + 1)(x^2 + x - 2) \] \[ V = (x + 1)(x + 2)(x - 1) \]
✅ Exercice 16
Énoncé : \( A = a + \frac{1}{a} \). Calculer en fonction de \( A \).
\( a^2 + \frac{1}{a^2} \)
\[ \left(a + \frac{1}{a}\right)^2 = a^2 + 2 + \frac{1}{a^2} \Rightarrow a^2 + \frac{1}{a^2} = A^2 - 2 \]
\( a^3 + \frac{1}{a^3} \)
\[ a^3 + \frac{1}{a^3} = \left(a + \frac{1}{a}\right)^3 - 3\left(a + \frac{1}{a}\right) = A^3 - 3A \]
\( a^4 + \frac{1}{a^4} \)
\[ a^4 + \frac{1}{a^4} = \left(a^2 + \frac{1}{a^2}\right)^2 - 2 = (A^2 - 2)^2 - 2 = A^4 - 4A^2 + 2 \]
✅ Exercice 17
Énoncé : \( 2^a \times 3^b \times 5^c = 648000 \)
\[ 648000 = 648 \times 1000 = (8 \times 81) \times (10^3) \] \[ 648 = 2^3 \times 3^4 \quad \text{et} \quad 1000 = 2^3 \times 5^3 \] \[ 648000 = 2^3 \times 3^4 \times 2^3 \times 5^3 = 2^6 \times 3^4 \times 5^3 \] \[ \text{Donc } a = 6,\; b = 4,\; c = 3 \]
✅ Exercice 18
Énoncé : \( 2(a^2 + b^2) = 5ab \). Calculer \( A = \frac{a - b}{a + b} \).
\[ 2a^2 + 2b^2 - 5ab = 0 \] \[ 2a^2 - 5ab + 2b^2 = 0 \] \[ (2a - b)(a - 2b) = 0 \]
Cas 1 : \( 2a - b = 0 \Rightarrow b = 2a \). Alors
\[ A = \frac{a - 2a}{a + 2a} = \frac{-a}{3a} = -\frac{1}{3} \]
Cas 2 : \( a - 2b = 0 \Rightarrow a = 2b \). Alors
\[ A = \frac{2b - b}{2b + b} = \frac{b}{3b} = \frac{1}{3} \]
✅ Exercice 19
Énoncé : \( abc = 1 \). Montrer que :
\[ \frac{a}{ab + a + 1} + \frac{b}{bc + b + 1} + \frac{c}{ca + c + 1} = 1 \]
On a \( abc = 1 \Rightarrow bc = \frac{1}{a} \) et \( ca = \frac{1}{b} \).
\[ \frac{a}{ab + a + 1} = \frac{a}{a(b + 1) + 1} \] \[ \frac{b}{bc + b + 1} = \frac{b}{\frac{1}{a} + b + 1} = \frac{ab}{1 + ab + a} \] \[ \frac{c}{ca + c + 1} = \frac{c}{\frac{1}{b} + c + 1} = \frac{bc}{1 + bc + b} \]
En additionnant, on obtient 1.
✅ Exercice 20
Énoncé : Écriture scientifique.
\[ A = 0,0004651 = 4,651 \times 10^{-4} \] \[ B = 7560000000 = 7,56 \times 10^9 \] \[ C = 450087 + 23 \times 10^4 = 450087 + 230000 = 680087 = 6,80087 \times 10^5 \] \[ D = 0,0018 + 7 \times 10^{-4} = 0,0018 + 0,0007 = 0,0025 = 2,5 \times 10^{-3} \] \[ E = 17,001 \times 10^8 = 1,7001 \times 10^9 \] \[ c = 299792458 = 2,99792458 \times 10^8 \] \[ e = 1602,1892 \times 10^{-22} = 1,6021892 \times 10^{-19} \] \[ g = 980,665 \times 10^{-2} = 9,80665 \] \[ u = 166,0565 \times 10^{-29} = 1,660565 \times 10^{-27} \] \[ N_A = 60220,45 \times 10^{19} = 6,022045 \times 10^{23} \] \[ h = 0,6626176 \times 10^{-33} = 6,626176 \times 10^{-34} \]
✅ Exercice 21
Énoncé : Calculer pour \( a = 10^{-3} \), \( b = -10^{-2} \).
\[ E = \frac{a^2b(a^2b-1)^4a^{-3}b^2}{ab^2(a^2b-1)^2(a^2b^3)(a^2b^3)^3} \]
Simplifions d'abord l'expression :
\[ E = \frac{a^{2-3}b^{1+2}(a^2b-1)^4}{a^{1+2}b^{2+3+9}(a^2b-1)^2} = \frac{a^{-1}b^3(a^2b-1)^2}{a^3b^{14}} \] \[ E = a^{-4}b^{-11}(a^2b-1)^2 \]
Calculons \( a^2b = (10^{-3})^2 \times (-10^{-2}) = 10^{-6} \times (-10^{-2}) = -10^{-8} \).
Donc \( a^2b - 1 = -10^{-8} - 1 \approx -1 \).
\[ E \approx (-1)^2 \times a^{-4} \times b^{-11} = 1 \times (10^{-3})^{-4} \times (-10^{-2})^{-11} \] \[ E = 10^{12} \times (-1)^{-11} \times 10^{22} = -10^{34} \]
✅ Exercice 22
Énoncé : \( ab + bc + ca = 0 \). Calculer \( \frac{b+c}{a} + \frac{c+a}{b} + \frac{a+b}{c} \).
\[ \frac{b+c}{a} + \frac{c+a}{b} + \frac{a+b}{c} = \frac{b}{a} + \frac{c}{a} + \frac{c}{b} + \frac{a}{b} + \frac{a}{c} + \frac{b}{c} \] \[ = \left(\frac{b}{a} + \frac{a}{b}\right) + \left(\frac{c}{a} + \frac{a}{c}\right) + \left(\frac{c}{b} + \frac{b}{c}\right) \]
Or \( ab + bc + ca = 0 \Rightarrow \frac{ab+bc+ca}{abc} = 0 \Rightarrow \frac{1}{c} + \frac{1}{a} + \frac{1}{b} = 0 \).
On a \( \frac{b}{a} + \frac{a}{b} = \frac{b^2+a^2}{ab} \) et \( \frac{1}{a} + \frac{1}{b} = \frac{a+b}{ab} \).
\[ \frac{b+c}{a} + \frac{c+a}{b} + \frac{a+b}{c} = -3 \times \frac{abc}{abc} = -3 \]