Question

1. Resolva as equações exponenciais:

a)2ˣ⁺³= 1/8 =
b) 5³ˣ⁺¹ +25
c) 81ˣ⁻² =⁴√27 =
d) √4ˣ⁺¹ = ³√16 =

Answer 1

Explicação passo a passo:

a)

$\sf{2^{x + 3} = \dfrac{1}{8}}$

$\sf{2^{x + 3} = 8^{-1}}$

$\sf{2^{x + 3} = (2^3)^{-1}}$

$\sf{2^{x + 3} = 2^{3 \times -1}}$

$\sf{2^{x + 3} = 2^{-3}}$

$\sf{\backslash \!\!\! 2^{x + 3} = \backslash \!\!\! 2^{-3}}$

$\sf{x + 3 = - 3}$

$\sf{x = - 3 - 3}$

$\green{\boxed{\sf{x = - 6}}}$

b)

$\sf{5^{3x + 1} = 25}$

$\sf{5^{3x + 1} = 5^2}$

$\sf{\backslash \!\!\! 5^{3x + 1} = \backslash \!\!\! 5^2}$

$\sf{3x + 1 = 2}$

$\sf{3x = 2 - 1}$

$\sf{3x = 1}$

$\green{\boxed{\sf{x = \dfrac{1}{3}}}}$

c)

$\sf{81^{x - 2} = \sqrt[4]{27}}$

$\sf{(3^4)^{x - 2} = 27^{\frac{1}{4}}}$

$\sf{3^{4 \times (x - 2)} = (3^3)^{\frac{1}{4}}}$

$\sf{3^{4x - 8} = 3^{\frac{3 \times 1}{4}}}$

$\sf{3^{4x - 8} = 3^{\frac{3}{4}}}$

$\sf{\backslash \!\!\! 3^{4x - 8} = \backslash \!\!\! 3^{\frac{3}{4}}}$

$\sf{4x - 8 = \dfrac{3}{4}}$

$\sf{4x = \dfrac{3}{4} + 8}$

$\sf{4x = \dfrac{8 \times 4 + 3}{4}}$

$\sf{4x = \dfrac{35}{4}}$

$\sf{x = \dfrac{35}{\dfrac{4}{4}}}$

$\sf{x = \dfrac{35}{4 \times 4}}$

$\green{\boxed{\sf{x = \dfrac{35}{16}}}}$

d)

$\sf{\sqrt{4^{x + 1}} = \sqrt[3]{16}}$

$\sf{4^{\frac{x + 1}{2}} = 16^{\frac{1}{3}}}$

$\sf{4^{\frac{x + 1}{2}} = (4^2)^{\frac{1}{3}}}$

$\sf{4^{\frac{x + 1}{2}} = 4^{\frac{2 \times 1}{3}}}$

$\sf{4^{\frac{x + 1}{2}} = 4^{\frac{2}{3}}}$

$\sf{\backslash \!\!\! 4^{\frac{x + 1}{2}} = \backslash \!\!\! 4^{\frac{2}{3}}}$

$\sf{\dfrac{x + 1}{2} = \dfrac{2}{3}}$

$\sf{(x + 1) \cdot 3 = 2 \cdot 2}$

$\sf{3x + 3 = 4}$

$\sf{3x = 4 - 3}$

$\sf{3x = 1}$

$\green{\boxed{\sf{x = \dfrac{1}{3}}}}$