L3 CalcLCC

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Universidade Federal da Paraba - UFPB
Departamento de Cincias Exatas - DCE
3a Lista de Exerc´ıcios - Ca´lculo 1
Prof. Carlos Alberto Gomes de Almeida
1. Calcule:
(a) 3
√
27 (b) 3
√
−27 (c) 13
√
−1 (d) 5
√
−32 (e) 31
√
0
2. Decomponha 729 em fatores primos, e extraia a raiz sexta de 729, isto e´, calcule 6
√
729.
3. Resolva a equac¸a˜o na inco´gnita x, em cada caso:
(a) (2x− 1)2 = 25 (b) (x+ 2)2 = a2 (c) (3x− 1)2 = (x+ 2)2
4. Verifique que:
(a) a3 − b3 = (a− b)(a2 + ab+ b2)
(b) a− b = ( 3
√
a− 3
√
b)(
3
√
a2 + 3
√
ab+
3
√
b2)
(c) x− y = (
√
x−
√
y)(
√
x+
√
y)
(d) x− y = ( 4
√
x− 4
√
y)(
4
√
x3 + 4
√
xy2 + 4
√
y3)
5. Racionalize:
(a)
3√
3
(b)
3
3
√
3
(c)
1√
5−
√
3
(d)
2√
11−
√
5
(e)
2√
11− 1
(f)
1
3
√
4− 3
√
2
(g) −
5
3
√
3− 2
(h)
1
4
√
x− 4
√
a
6. Considere o polinoˆmio do 20 grau ax2 + bx+ c, onde a,b, e c ∈ R com a 6= 0
(a) Verifique que: ax2 + bx+ c = a
[(
x+
b
2a
)2
−
∆
4a2
]
, ∆ = b2 − 4ac
(b) Conclua de (a) que, se ∆ > 0, as ra´ızes de ax2 + bx+ c sa˜o dadas pela fo´rmula x = −b±
√
∆
2a
7. Simplifique:
(a)
24
√
412 (b)
45
√
845 (c)
15
√
232 (d)
28
√
232 (e) 3
√
a
√
a
8. Simplifique:
(a) x 3
√
x+ 4x4/3 − 5
3
√
x4
(b)
3
√
x2
√
x3 − 2x2 6
√
x
6
√
x13
(c)
5
√
xx2x1/3 − (
15
√
x2)2x
15
√
x19
(d) (
3
√
5a2/3)9
(e)
4
√
3 3
√
3
3
√
3
1
9. Simplifique:
(a) 3
√
1024
(b) 5
√
15552
(c)
√
1800
(d) 3
√
56
(e) 4
√
14256
(f)
32 3
√
3
√
2√
512
(g)
3
√
a4b7
(h)
3
√
a10b6
4
√
a2b5
10. Simplifique
(a)
√
4+ 2
√
3 (b)
√
9+ 2
√
14 (c)
√
15+
√
176 (d)
√
11− 2
√
18
11. Determine dois nu´meros cuja soma e´
√
2 e cujo o produto e´ −6− 2
√
2.
12. Resolva a equac¸a˜o, em cada caso:
(a)
√
2x+ 4 = 10− x
(b)
√
2x+ 4 = x− 10
(c) 4
√
x− 2 = x+ 1
(d) 2
√
x+ x = 3
(e)
√
1+ 4x = 3+
√
x− 2
(f)
√
1+ 3x−
√
4− x = 1
(g)
√
3+ 2x−
√
x− 2+
√
1+ x = 0
13. Resolva a equac¸a˜o (x+
1
x
)2 − (x+
1
x
) − 2 = 0
BONS ESTUDOS!!!
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