Equação Exponencial 1.docx
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1) Resolver a equação exponencial abaixo 𝑋
𝑋
(√5 + 2√6) + (√5 − 2√6) = 10 𝑋
𝑎 = (√5 + 2√6)
√5+2√6 Segue que√5 − 2√6 = √5 − 2√6. √5+2√6
√52 −(2√6)2
√5 − 2√6=
√5+2√6
=
1 √5+2√6 𝑋
𝑋
1
(√5 − 2√6) = (
√5 + 2√6
)
1
𝑎 + = 10
A equação fica:
𝑎
𝑎2 + 1 = 10𝑎 →
𝑎2 − 10𝑎 + 1 = 0
∆= (−10)2 − 4.1.1= 100 – 4 = 96 → √∆=√96=4√6
𝑎=
10 ± 4√6 = 5 ± 2√6 2 𝑋
1)
(√5 + 2√6) = 5 + 2√6
→X=2
2)
(√5 + 2√6) = 5 − 2√6 =
1 = 5+2√6
Conjunto solução: S = { -2, 2}
−1
(5 + 2√6) → X = -2
2) Resolver a equação exponencial X X XX
49
7
= 77
76
7
7
= √7 X
=7
X 49
√7
6
76
7−1 .77
= (7
49 7
7
= √7 7
7
= √7
√7
√7
77 7−1
)
49
= 77
= 7−1 . 77
=
7 1 7 (77 )
49 7
= ( √7)
49
49
X = √7 7
Elevando a 49 ambos os lados 49 49
( 𝑥𝑥 ) (𝑥 49 )𝑥
49
6 49
= (77 )
= (77 )7
𝑥 49 = 77 𝑥 = (7
1 7 )49
1
7
𝑥 = 77 = √7
7
6
6
= (749 )7 = (77.7 )7
6
77
7 7
= ( √7)
√749
3) Se X X
49
6
−4
= 77 e y √y = √4
Calcule S= X 7 + 8√y a) Elevando a 49 ambos os membros XX
49
= 77
6 49
49 49
( 𝑥𝑥 ) (𝑥 49 )𝑥
49
6
= (77 )
= (77 )7
6
= (749 )7 = (77.7 )7
6
7
𝑥 49 = 77 (𝑥 7 )7 = 77 7
𝑥 7 = √7 7 𝑥7 = 7 −4
b) y √y = √4 8
4
−4
1
16
−4
√4 = √4−1 = √4−2 = √4−4 = (4−4 )16 = (4−4 )2
Extraindo a raiz quadrada em ambos os lados √y √y √y
√y
−4
= √ √4 =
√(4−4 )2−4
−4
= (2−4 )2
√y = 2−4 y = (2−4 )2 = 2−8 4
4
√y = √√y = √2−4 = 2−1
8
8
√y =
1 2
S = X 7 + 8√y = 7 +
1 = 7,5 2
=
√4−4
2−4
𝑋
(√5 + 2√6) + (√5 − 2√6) = 10 𝑋
𝑎 = (√5 + 2√6)
√5+2√6 Segue que√5 − 2√6 = √5 − 2√6. √5+2√6
√52 −(2√6)2
√5 − 2√6=
√5+2√6
=
1 √5+2√6 𝑋
𝑋
1
(√5 − 2√6) = (
√5 + 2√6
)
1
𝑎 + = 10
A equação fica:
𝑎
𝑎2 + 1 = 10𝑎 →
𝑎2 − 10𝑎 + 1 = 0
∆= (−10)2 − 4.1.1= 100 – 4 = 96 → √∆=√96=4√6
𝑎=
10 ± 4√6 = 5 ± 2√6 2 𝑋
1)
(√5 + 2√6) = 5 + 2√6
→X=2
2)
(√5 + 2√6) = 5 − 2√6 =
1 = 5+2√6
Conjunto solução: S = { -2, 2}
−1
(5 + 2√6) → X = -2
2) Resolver a equação exponencial X X XX
49
7
= 77
76
7
7
= √7 X
=7
X 49
√7
6
76
7−1 .77
= (7
49 7
7
= √7 7
7
= √7
√7
√7
77 7−1
)
49
= 77
= 7−1 . 77
=
7 1 7 (77 )
49 7
= ( √7)
49
49
X = √7 7
Elevando a 49 ambos os lados 49 49
( 𝑥𝑥 ) (𝑥 49 )𝑥
49
6 49
= (77 )
= (77 )7
𝑥 49 = 77 𝑥 = (7
1 7 )49
1
7
𝑥 = 77 = √7
7
6
6
= (749 )7 = (77.7 )7
6
77
7 7
= ( √7)
√749
3) Se X X
49
6
−4
= 77 e y √y = √4
Calcule S= X 7 + 8√y a) Elevando a 49 ambos os membros XX
49
= 77
6 49
49 49
( 𝑥𝑥 ) (𝑥 49 )𝑥
49
6
= (77 )
= (77 )7
6
= (749 )7 = (77.7 )7
6
7
𝑥 49 = 77 (𝑥 7 )7 = 77 7
𝑥 7 = √7 7 𝑥7 = 7 −4
b) y √y = √4 8
4
−4
1
16
−4
√4 = √4−1 = √4−2 = √4−4 = (4−4 )16 = (4−4 )2
Extraindo a raiz quadrada em ambos os lados √y √y √y
√y
−4
= √ √4 =
√(4−4 )2−4
−4
= (2−4 )2
√y = 2−4 y = (2−4 )2 = 2−8 4
4
√y = √√y = √2−4 = 2−1
8
8
√y =
1 2
S = X 7 + 8√y = 7 +
1 = 7,5 2
=
√4−4
2−4
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