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Apuntes

``` 1. a) \(\frac{(2x^3)^2(3x^4)}{(x^3)^4} = \frac{(4x^6)(3x^4)}{x^{12}}\) = \(\frac{12x^{10}}{x^{12}}\) = \(12x^{10-12}\) = \(12x^{-2}\) = \(\frac{12}{x^2}\) b) \((6^4) \left(\frac{1}{3}\right)^2 (125^6) = \frac{12}{3} 6^{4+2-8}\) = \(4b^{-2}\) = \(\frac{4}{b^2}\) 2. a) \(\frac{3}{x^4 y^6} \cdot \frac{1}{3} \cdot \frac{8}{x y}\) = \(\frac{3}{x^4 y^6} \cdot \frac{8}{3x y}\) = \(\frac{3}{3} \cdot \frac{8}{x^4 y^6} \cdot \frac{1}{x y}\) = \(\frac{2x y^3}{3x^2 y}\) 3. a) \((2x+3)(2x-3)\) = \((2x)^2-3^2\) = \(4x^2-9\) b) \((2x^2-3y^2)^2 = (2x)^2 - 2(2x)(3y^2) + (3y^2)^2\) = \(4x^2 - 12x^2y^4 + 9y^4\) c) \((4-3x)^3\) = \(4^3 - 3(4)^2(3x) + 3(4)(3x)^2 - (3x)^2\) = \(64 - 9(16)x + 12(9x^2) - 27x^3\) = \(64 - 144x + 108x^2 - 27x^3\) 4. \(5(3x^2 + 6x^2 - 4x + 7) - (3x^2 - 2x - 6)\) = \(5x^3 + 30x^2 - 20x + 35 - 3x^2 + 2x + 6\) = \(5x^3 + 27x^2 - 18x + 41\) 5. a) \(y^2 + 14y + 49 = (y+7)^2\) \(2(y)(19)\) b) \(8x - 125 = (2x)^3 - 5^3\) = \((2x-5)((2x)^2 + 2x(5) + 5^2)\) = \((2x-5)(4x^2 + 10x + 25)\) c) \(x^2 - 2x - 8 = (x-4)(x+2)\) 6. \(4(3x-4) = 5(x-2)\) \(5(x-2) - 14x = x - 5\) \(5x - 10 - 14x = x - 5\) \(-9x - x...