Réponse :
1) a) 27² = ?³ ⇔ ? = ∛27² = 9
27² = 9³
b) (12⁵)⁴ = ?⁵ ⇔ (12⁴)⁵ = ?⁵ ⇔ ? = 12⁴ = 20736
(12⁵)⁴ = 20736⁵
c) 36⁻¹ = (1/6)^? ⇔ 36⁻¹ = (1/36)²
d) 25¹⁰ = ?⁴ ⇔ (5²)¹⁰ = ?⁴ ⇔ 5²⁰ = ?⁴ ⇔ (5⁵)⁴ = ?⁴ ⇔ ? = 5⁵ = 3125
25¹⁰ = 3125⁴
2) développements
a) (x + y + 1)(x + y) = x² + x y + y x + y² + x + y = x² + 2 x y + x + y + y²
b) (x²+2)(x - 2) = x³ - 2 x² + 2 x - 4
c) (x²+1)(x² - 1) = x⁴ - 1
d) (x² + x + 1)(x + 2) = x³ + 2 x² + x² + 2 x + x + 2 = x³ + 3 x² + 3 x + 2
3) factorisations
a) (x² + 2)(x - 3) + (2 - x)(x²+2) = (x²+2)(x - 3 + 2 - x) = - (x² + 2)
b) 8(x+3+y)+4(x+3+y)-10(x+3+y) = 2(x+3+y)
c) x⁴ + x² + x = x(x³ + x + 1)
d) (6 - x)(x + 1) - (x - 6)(x + 2) = (6 - x)(x + 1 - x - 2) = (6 - x)(- 1) = (x - 6)
4) Equations
a) 3(x + 3) + 2(x - 2) = 2 x + 5 ⇔ 3 x + 9 + 2 x - 4 = 2 x + 5
⇔ 5 x + 5 = 2 x + 5 ⇔ 3 x = 0 ⇔ x = 0
b) 4 x + 2 - 5 x = 4 - 5 x ⇔ - x + 2 = 4 - 5 x ⇔ 4 x = 2 ⇔ x = 2/4 = 1/2
c) 50 x + 75 x = 25 x + 50 ⇔ 125 x = 25 x + 50 ⇔ 100 x = 50 ⇔ x = 50/100 = 1/2
d) (x+2)(x - 3) + (x+2)(4 - x) = 3 ⇔ (x + 2)(x - 3 + 4 - x) = 3
⇔ x + 2 = 3 ⇔ x = 1
Explications étape par étape