Question

Resolva as equações e encontre as soluções:

B)
$( {3}^{x} ) {}^{2x - 7} = \frac{1}{27}$
C)
$( {3}^{x} ) {}^{x } = 19683$
D)
${4}^{x \: 2 + x - 12} = 1$
E)
$( {10}^{x} ) {}^{x - 4} = \frac{1}{100}$
F)
$( {3}^{ - x} ) {}^{x - 5} = 81$
​

Answer 1

b) (3ˣ)²ˣ⁻⁷ = 1/27

(3ˣ)²ˣ⁻⁷ = 1/3³

(3ˣ)²ˣ⁻⁷ = 3⁻³

2x² - 7x = - 3

2 x² - 7x + 3 = 0

Δ = b² - 4ac

Δ = (-7)² - 4(2)(3)

Δ = 49 - 24

Δ = 25

x = - b ± √Δ / 2a

x = 7 ± 5 / 2(2)

x' = 12/4

x' = 3

x'' = 2/4

x'' = 1/2

S = { 1/2, 3 }

c) (3ˣ)ˣ = 19 683

3ˣ² = 3⁹

x² = 9

x = ± √9

x = ± 3

S = { -3, 3 }

d) (4ˣ)²⁺ˣ⁻¹² = 1

(4ˣ)ˣ⁻¹⁰ = 4⁰

4ˣ²⁻¹⁰ˣ = 4⁰

x² - 10x = 0

x(x - 10) = 0

x = 0

ou

x - 10 = 0

x = 10

S = { 0, 10 }

e) (10ˣ)ˣ⁻⁴ = 1/100

(10ˣ)ˣ⁻⁴ = 1/10²

(10ˣ)ˣ⁻⁴ = 10⁻²

10ˣ²⁻⁴ˣ = 10⁻²

x² - 4x = - 2

x² - 4x + 2 = 0

Δ = b² - 4ac

Δ = (-4)² - 4(1)(2)

Δ = 16 - 8

Δ = 8

x = - b ± √Δ / 2a

x = 4 ± √8 / 2(1)

x = 4 ± 2√2 / 2

x' = 4 + 2√2 / 2

x' = 2 + √2

x'' = 4 - 2√2 / 2

x'' = 2 - √2

S = { 2 - √2,  2 + √2  }

f) (3⁻ˣ)ˣ⁻⁵ = 81

(3⁻ˣ)ˣ⁻⁵ = 3⁴

3⁻ˣ²+⁵ˣ = 3⁴

- x² + 5x = 4

- x² + 5x - 4 = 0

Δ = b² - 4ac

Δ = (5)² - 4(-1)(-4)

Δ = 25 - 16

Δ = 9

x = - b ± √Δ / 2a

x =  - 5 ± 3 / 2(-1)

x' = - 5 + 3 / - 2

x' = - 2 / - 2

x' = 1

x'' = - 5 - 3 / - 2

x'' =  - 8 / - 2

x'' = 4

S = { 1, 4 }