sábado, 4 de maio de 2013

Exercícios Resolvidos de Matemática 01


      Exercícios

1)      3 + (-5) + (-9) + 2 = 3 – 5 – 9 + 2 = - 9
2)      -18 – 8 – 6 + 14 + 20 + 50 – 2 = 50
3)      -2(8+3) + 3(-5) = -16 – 6 – 15 = -37
4)      3¯⁴ = 1/3⁴ = 1/81
5)      7,4⁰ = 1
6)      4² (4¯² + 2²) = 16(1/4² + 4) = 16(1/16 + 4) = 16/16 + 64 = 1 + 64 = 65
7)      1/6¯¹ - (3²/6²) = 1/(1/6) – (9/36) = 6 – 1/4 = (24 – 1)/4 = 23/4

Calcula

1)      -3z²(-4z²+2g) = 12z⁴-6gz²
2)      5y²(y-3) – 6(2y² - 3y) = 5yᶟ - 15y² - 12y² + 18y = 5yᶟ - 27y² + 18y
3)      (4p – 3)(2p + 1) – (p + 5)(p – 1) = 8p² + 4p – 6p -3 – (p²- p +5p – 5) =
= 8p² + 4p – 6p – 3 - p² + p – 5p + 5 = 8p² - p² + (4 – 6 + 1 – 5)p – 3 + 5 = 7p² - 6p + 2

Efectua

1)      Y – (4x – 2xy) – [4y – (-3x + 8y – 2xy)] = y – 4x +2xy – [ 4y + 3x – 8y + 2xy] =

= y - 4x + 2xy – 4y – 3x + 8y – 2xy = (1 – 4 + 8)y + (-4 -3)x = 5y – 7x

2)      5 – [ -(25¯¹ + 66 – 8) + (- 7b + 8b¯¹) – 12] – 5b¯¹ =
= 5 – [- (1/25 +58) + (-7b² + 8/b) – 12] – 5/b =
= 5 – [ -(1+1450)/(25) + (-7b² + 8)/(b) – 12] -5/b =
=  5 – [ -1451/25 + (-7b² + 8)/(b) – 12 ] – 5/b = 5 + 1451/25 – (-7b² + 8)/(b) + 12 – 5/b =
= 1451/25 – (-7b² + 8)/(b) – 5/b + 17 = (1451 + 425)/25 – (-7b² + 8 – 5)/(b) =
= 1876/25 – (-7b² + 3)/(b)

3)      2b² - { - 12 + 2 [ -5b² + (2b – 3)(b + 4) +9b} – 6 =
= 2b² - { -12 + 2 [ -5b² +2b² + 8b -3b -12] + 9b -6 =
= 2b² - { -12 + 2 [-3b² + 5b – 12] + 9b} – 6 = 2b² - { -12 – 6b² + 10b – 24 + 9b } – 6 =
= 2b² - {-6b² + 19b – 36} – 6 = 2b² + 6b² - 19b + 36 -6 = 8b² - 19b + 30

Calcula

1)      ⁴√81 = ⁴√3⁴ = 3
2)      ⁵√-32 = ⁵√2⁵ = 2
3)      ᶟ√1/8 = ᶟ√1/2ᶟ = ᶟ√1/ᶟ√2ᶟ = ½
4)      ⁶√1/64 = ⁶√2⁶ = 2
5)      - ᶟ√-125 = - ᶟ√-5ᶟ = 5

 Determine M.M.C (minimo multiplo comum) e M.D.C (máximo divisor comum)

1)      18 e 24 = 72 (m.m.c) e 6 (m.d.c)
Demonstração: decompondo em factores primos teremos
18 =  2 x  3 x 3 = 2 x 3²
24 = 2 x 2 x 2 x 3 = 2ᶟ x 3
Daí pegamos os factores primos comuns e não comuns de ambos números (2 e 3), e multiplicamos os de maior expoente (2ᶟ x 3² = 72)

2)      15 ; 20 ; 25 = 150 (m.m.c) 5 (m.d.c)
15 = 3 x 5
20 = 2 x 2 x 5 = x 5
25 = 5 x 5 =
m.m.c = 3 x 2² x 5² = 150 
3)      12a⁵b²; 21a⁵b⁶ = 84 (m.m.c) e 3 (m.d.c)

Efectua

a)      2x + 3(4x – 8) = 2x +12x – 24 = 14x -24
b)      3(4a – 1) - 2(2a – 5) = 12a – 3 – 4a + 10 = 8a + 7
c)       (m + 3)(m – 5) + 3m(-2m + 6) = m² - 5m + 3m – 15 – 6 m² + 18m = - 5m² + 16m – 15
d)      (6m + 5)² + 2(5 – 3m)² = 36m² + 2(6*5)m + 25 +2(25- 2(5*3m) + 9m²) =
= 36m² + 60m + 25 +2(25 – 30m + 9m²) = 36m² + 60m +25 +50 - 60m + 18m² =
= 54m² + 75
e)      (a + 5)(a – 5) + (a + 2)² = a² - 2(a*5) + 25 + a² + 2(a*2) +4 = a² -10a + 25 + a² + 4a + 4 =
= 2a² - 6a + 29

Decompõe

1)      12a⁵ - 4aᶟ = 2²*3*a⁵ - 2²aᶟ = 2² (3a⁵ - aᶟ) = 4 (3a⁵ - aᶟ)
2)      24xᶟy⁴ - 18x⁴yᶟ - 30x⁵y² = 2(12xᶟy⁴ - 9x⁴yᶟ - 15x⁵y²)
3)      16b² - 81c² = (4b)² - (9c)² = (4b + 9c)(4b – 9c)
4)      (1/9)m⁴ - 100a⁴ = (1/3²)m⁴ - 10²a⁴ = (m²)²/3² - (10a²)² = [(m²)/3 + 10a²][(m²)/3 – 10a²]
5)      4x² - 9y² = 2²x² - 3²y² = (2x)² - (3y)² = (2x – 3y)(2x + 3y)
6)      4xᶟ - 49x = 4x(x² - 7)
7)      x² - 10x +25 = (x – 5)(x – 5) = (x – 5)²
8)      c² + 18c + 81 = (c + 9)(c + 9) = (c + 9)²
9)      4b² - 12b +9 = (2b – 3)(2b – 3) = (2b – 3)²
10)   c² - 10c + 24 = (c – 6)(c – 4)
11)   x² + 7x + 10 = (x + 2)(x + 5)