Question

Resolva as equações:

a) $3^{x+1}= 27$
b) $(0,25)^{2x+3} =4^{x-1}$
c) $\sqrt{7^{x} } = 343$
d) $5^{\frac{3}{2} x } = 125^{4}$
e) $\sqrt[6]{1000} . 10^{x} = 0,001$
f) $\frac{8^{x} }{2^{x+2} } = 256$

Eu queria todos os cálculos, por gentileza... ;-;
S.O.C.O.R.R.O.

Answer 1

a) $3^{x+1}=27$

$3^{x+1}=3^3$

$x+1=3$

$x=3-1$

$x=2$

b) $(0,25)^{2x+3}=4^{x-1}$

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

$(4^{-1})^{2x+3}=4^{x-1}$

$4^{-2x-3}=4^{x-1}$

$-2x-3=x-1$

$-2x-x=3-1$

$-3x=2$

$x=-\frac{2}{3}$

c) $\sqrt{7^x}=343$

$7^{\frac{x}{2} }=7^3$

$\frac{x}{2}=3$

$x=2\cdot 3$

$x=6$

d) $5^{\frac{3}{2}x }=125^4$

$5^{\frac{3x}{2} }=(5^3)^4$

$5^{\frac{3x}{2} }=5^{12}$

$\frac{3x}{2}=12$

$3x=12\cdot 2$

$3x=24$

$x=\frac{24}{3}$

$x=8$

e) $\sqrt[6]{1000}\cdot 10^x=0,001$

$\sqrt[6]{10^3}\cdot 10^x=10^{-3}$

$10^{\frac{3}{6} }\cdot 10^x=10^{-3}$

$10^{x+\frac{3}{6} }=10^{-3}$

$x+\frac{3}{6}=-3$

$x+\frac{1}{2}=-3$

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

$x=-\frac{6}{2}-\frac{1}{2}$

$x=-\frac{7}{2}$

f) $\frac{8^x}{2^{x+2}}=256$

$\frac{(2^3)^x}{2^{x+2}}=2^8$

$\frac{2^{3x}}{2^{x+2}}=2^8$

$2^{3x-(x+2)}}=2^8$

$3x-(x+2)=8$

$3x-x-2=8$

$3x-x=8+2$

$2x=10$

$x=\frac{10}{2}$

$x=5$