Question 92970
{{{2(4^(x+1)) = 1}}}


{{{4^(x+1) = 1/2}}} Divide both sides by 2


{{{((1/2)^(-2))^(x+1) = 1/2}}} Rewrite 4 as {{{(1/2)^(-2)}}}


{{{(1/2)^(-2(x+1)) = 1/2}}} Multiply the exponents



{{{(1/2)^(-2x-2) = 1/2}}} Distribute


{{{(1/2)^(-2x-2) = (1/2)^1}}} Rewrite {{{1/2}}} as {{{(1/2)^1}}}



Now that we have the same base, we can set the exponents equal to each other



{{{-2x=1+2}}} Add 2 to both sides





{{{-2 x=3}}} Combine like terms 2 and 1 on the right side to get 3






{{{x=(3)/(-2)}}} Now divide both sides by  -2 to isolate and solve for x



{{{x= -3 / 2}}} Reduce




So our answer is

{{{x= -3 / 2}}}


========================================================================

Check:


{{{2(4^(x+1)) = 1}}} Start with the given equation


{{{2(4^((-3/2)+1)) = 1}}} Plug in {{{x=-3 / 2}}}


{{{2(4^(-1/2)) = 1}}} Combine like terms


{{{2(sqrt(4^-1)) = 1}}} Rewrite {{{4^(-1/2)}}} as {{{sqrt(4^-1)}}}


{{{2(sqrt(1/4)) = 1}}} Evaluate {{{4^-1}}} to get  {{{1/4}}}



{{{2(1/2) = 1}}} Take the square root of {{{1/4}}} to get {{{1/2}}}



{{{1 = 1}}} Multiply. Since the two equations are equal, this verifies our answer.